*273*
*54*
*21MB*

*English*
*Pages [802]*
*Year 2016*

Deep Learning Ian Goodfellow Yoshua Bengio Aaron Courville

Contents Website

vii

Acknowledgments

viii

Notation

xi

1

Introduction 1.1 Who Should Read This Book? . . . . . . . . . . . . . . . . . . . . 1.2 Historical Trends in Deep Learning . . . . . . . . . . . . . . . . .

1 8 11

I

Applied Math and Machine Learning Basics

29

2

Linear Algebra 2.1 Scalars, Vectors, Matrices and Tensors . . 2.2 Multiplying Matrices and Vectors . . . . . 2.3 Identity and Inverse Matrices . . . . . . . 2.4 Linear Dependence and Span . . . . . . . 2.5 Norms . . . . . . . . . . . . . . . . . . . . 2.6 Special Kinds of Matrices and Vectors . . 2.7 Eigendecomposition . . . . . . . . . . . . . 2.8 Singular Value Decomposition . . . . . . . 2.9 The Moore-Penrose Pseudoinverse . . . . . 2.10 The Trace Operator . . . . . . . . . . . . 2.11 The Determinant . . . . . . . . . . . . . . 2.12 Example: Principal Components Analysis

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31 31 34 36 37 39 40 42 44 45 46 47 48

Probability and Information Theory 3.1 Why Probability? . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 54

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CONTENTS

3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 4

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II 6

Random Variables . . . . . . . . . . . . . . Probability Distributions . . . . . . . . . . . Marginal Probability . . . . . . . . . . . . . Conditional Probability . . . . . . . . . . . The Chain Rule of Conditional Probabilities Independence and Conditional Independence Expectation, Variance and Covariance . . . Common Probability Distributions . . . . . Useful Properties of Common Functions . . Bayes’ Rule . . . . . . . . . . . . . . . . . . Technical Details of Continuous Variables . Information Theory . . . . . . . . . . . . . . Structured Probabilistic Models . . . . . . .

Numerical Computation 4.1 Overﬂow and Underﬂow . . . . 4.2 Poor Conditioning . . . . . . . 4.3 Gradient-Based Optimization . 4.4 Constrained Optimization . . . 4.5 Example: Linear Least Squares

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Machine Learning Basics 5.1 Learning Algorithms . . . . . . . . . . . 5.2 Capacity, Overﬁtting and Underﬁtting . 5.3 Hyperparameters and Validation Sets . . 5.4 Estimators, Bias and Variance . . . . . . 5.5 Maximum Likelihood Estimation . . . . 5.6 Bayesian Statistics . . . . . . . . . . . . 5.7 Supervised Learning Algorithms . . . . . 5.8 Unsupervised Learning Algorithms . . . 5.9 Stochastic Gradient Descent . . . . . . . 5.10 Building a Machine Learning Algorithm 5.11 Challenges Motivating Deep Learning . . Deep Networks: Modern Practices

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98 99 110 120 122 131 135 139 145 150 152 154 165

Deep Feedforward Networks 167 6.1 Example: Learning XOR . . . . . . . . . . . . . . . . . . . . . . . 170 6.2 Gradient-Based Learning . . . . . . . . . . . . . . . . . . . . . . . 176 ii

CONTENTS

6.3 6.4 6.5 6.6

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190 196 203 224

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Regularization for Deep Learning 7.1 Parameter Norm Penalties . . . . . . . . . . . . . . . . . . . . . . 7.2 Norm Penalties as Constrained Optimization . . . . . . . . . . . . 7.3 Regularization and Under-Constrained Problems . . . . . . . . . 7.4 Dataset Augmentation . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Noise Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Semi-Supervised Learning . . . . . . . . . . . . . . . . . . . . . . 7.7 Multi-Task Learning . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Early Stopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9 Parameter Tying and Parameter Sharing . . . . . . . . . . . . . . 7.10 Sparse Representations . . . . . . . . . . . . . . . . . . . . . . . . 7.11 Bagging and Other Ensemble Methods . . . . . . . . . . . . . . . 7.12 Dropout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.13 Adversarial Training . . . . . . . . . . . . . . . . . . . . . . . . . 7.14 Tangent Distance, Tangent Prop, and Manifold Tangent Classiﬁer

228 230 237 239 240 242 243 244 246 253 254 256 258 268 270

8

Optimization for Training Deep Models 8.1 How Learning Diﬀers from Pure Optimization 8.2 Challenges in Neural Network Optimization . 8.3 Basic Algorithms . . . . . . . . . . . . . . . . 8.4 Parameter Initialization Strategies . . . . . . 8.5 Algorithms with Adaptive Learning Rates . . 8.6 Approximate Second-Order Methods . . . . . 8.7 Optimization Strategies and Meta-Algorithms

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274 275 282 294 301 306 310 317

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330 331 335 339 345 347 358 360 362 363

9

Hidden Units . . . . . . . . . . . . . . . . . . . . . . . . Architecture Design . . . . . . . . . . . . . . . . . . . . . Back-Propagation and Other Diﬀerentiation Algorithms Historical Notes . . . . . . . . . . . . . . . . . . . . . . .

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Convolutional Networks 9.1 The Convolution Operation . . . . . . . . . . . . . . . 9.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Pooling . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Convolution and Pooling as an Inﬁnitely Strong Prior . 9.5 Variants of the Basic Convolution Function . . . . . . 9.6 Structured Outputs . . . . . . . . . . . . . . . . . . . . 9.7 Data Types . . . . . . . . . . . . . . . . . . . . . . . . 9.8 Eﬃcient Convolution Algorithms . . . . . . . . . . . . 9.9 Random or Unsupervised Features . . . . . . . . . . . iii

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CONTENTS

9.10 The Neuroscientiﬁc Basis for Convolutional Networks . . . . . . . 364 9.11 Convolutional Networks and the History of Deep Learning . . . . 371 10 Sequence Modeling: Recurrent and Recursive Nets 10.1 Unfolding Computational Graphs . . . . . . . . . . . . . . . 10.2 Recurrent Neural Networks . . . . . . . . . . . . . . . . . . 10.3 Bidirectional RNNs . . . . . . . . . . . . . . . . . . . . . . . 10.4 Encoder-Decoder Sequence-to-Sequence Architectures . . . . 10.5 Deep Recurrent Networks . . . . . . . . . . . . . . . . . . . 10.6 Recursive Neural Networks . . . . . . . . . . . . . . . . . . . 10.7 The Challenge of Long-Term Dependencies . . . . . . . . . . 10.8 Echo State Networks . . . . . . . . . . . . . . . . . . . . . . 10.9 Leaky Units and Other Strategies for Multiple Time Scales . 10.10 The Long Short-Term Memory and Other Gated RNNs . . . 10.11 Optimization for Long-Term Dependencies . . . . . . . . . . 10.12 Explicit Memory . . . . . . . . . . . . . . . . . . . . . . . . 11 Practical Methodology 11.1 Performance Metrics . . . . . . . . . . . . . 11.2 Default Baseline Models . . . . . . . . . . . 11.3 Determining Whether to Gather More Data 11.4 Selecting Hyperparameters . . . . . . . . . . 11.5 Debugging Strategies . . . . . . . . . . . . . 11.6 Example: Multi-Digit Number Recognition . 12 Applications 12.1 Large Scale Deep Learning . . 12.2 Computer Vision . . . . . . . 12.3 Speech Recognition . . . . . . 12.4 Natural Language Processing 12.5 Other Applications . . . . . . III

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373 375 378 395 396 398 400 402 405 408 410 414 418

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445 445 454 460 463 479

Deep Learning Research

488

13 Linear Factor Models 13.1 Probabilistic PCA and Factor Analysis . 13.2 Independent Component Analysis (ICA) 13.3 Slow Feature Analysis . . . . . . . . . . 13.4 Sparse Coding . . . . . . . . . . . . . . . iv

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491 492 493 495 498

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13.5 Manifold Interpretation of PCA . . . . . . . . . . . . . . . . . . . 501 14 Autoencoders 14.1 Undercomplete Autoencoders . . . . . . . . . 14.2 Regularized Autoencoders . . . . . . . . . . . 14.3 Representational Power, Layer Size and Depth 14.4 Stochastic Encoders and Decoders . . . . . . . 14.5 Denoising Autoencoders . . . . . . . . . . . . 14.6 Learning Manifolds with Autoencoders . . . . 14.7 Contractive Autoencoders . . . . . . . . . . . 14.8 Predictive Sparse Decomposition . . . . . . . 14.9 Applications of Autoencoders . . . . . . . . .

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15 Representation Learning 15.1 Greedy Layer-Wise Unsupervised Pretraining . . 15.2 Transfer Learning and Domain Adaptation . . . . 15.3 Semi-Supervised Disentangling of Causal Factors 15.4 Distributed Representation . . . . . . . . . . . . . 15.5 Exponential Gains from Depth . . . . . . . . . . 15.6 Providing Clues to Discover Underlying Causes .

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16 Structured Probabilistic Models for Deep Learning 16.1 The Challenge of Unstructured Modeling . . . . . . . . . 16.2 Using Graphs to Describe Model Structure . . . . . . . . 16.3 Sampling from Graphical Models . . . . . . . . . . . . . 16.4 Advantages of Structured Modeling . . . . . . . . . . . . 16.5 Learning about Dependencies . . . . . . . . . . . . . . . 16.6 Inference and Approximate Inference . . . . . . . . . . . 16.7 The Deep Learning Approach to Structured Probabilistic 17 Monte Carlo Methods 17.1 Sampling and Monte Carlo Methods . . . . . . . . 17.2 Importance Sampling . . . . . . . . . . . . . . . . . 17.3 Markov Chain Monte Carlo Methods . . . . . . . . 17.4 Gibbs Sampling . . . . . . . . . . . . . . . . . . . . 17.5 The Challenge of Mixing between Separated Modes

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504 505 506 510 511 512 517 523 525 526

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18 Confronting the Partition Function 607 18.1 The Log-Likelihood Gradient . . . . . . . . . . . . . . . . . . . . 608 18.2 Stochastic Maximum Likelihood and Contrastive Divergence . . . 609 v

CONTENTS

18.3 18.4 18.5 18.6 18.7

Pseudolikelihood . . . . . . . . . . . Score Matching and Ratio Matching Denoising Score Matching . . . . . . Noise-Contrastive Estimation . . . . Estimating the Partition Function . .

19 Approximate Inference 19.1 Inference as Optimization . . . . . 19.2 Expectation Maximization . . . . . 19.3 MAP Inference and Sparse Coding 19.4 Variational Inference and Learning 19.5 Learned Approximate Inference . .

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20 Deep Generative Models 20.1 Boltzmann Machines . . . . . . . . . . . . . . . . . . . . . 20.2 Restricted Boltzmann Machines . . . . . . . . . . . . . . . 20.3 Deep Belief Networks . . . . . . . . . . . . . . . . . . . . . 20.4 Deep Boltzmann Machines . . . . . . . . . . . . . . . . . . 20.5 Boltzmann Machines for Real-Valued Data . . . . . . . . . 20.6 Convolutional Boltzmann Machines . . . . . . . . . . . . . 20.7 Boltzmann Machines for Structured or Sequential Outputs 20.8 Other Boltzmann Machines . . . . . . . . . . . . . . . . . 20.9 Back-Propagation through Random Operations . . . . . . 20.10 Directed Generative Nets . . . . . . . . . . . . . . . . . . . 20.11 Drawing Samples from Autoencoders . . . . . . . . . . . . 20.12 Generative Stochastic Networks . . . . . . . . . . . . . . . 20.13 Other Generation Schemes . . . . . . . . . . . . . . . . . . 20.14 Evaluating Generative Models . . . . . . . . . . . . . . . . 20.15 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .

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617 619 621 622 625

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656 656 658 662 665 678 685 687 688 689 694 712 716 717 719 721

Bibliography

723

Index

780

vi

Website www.deeplearningbook.org

This book is accompanied by the above website. The website provides a variety of supplementary material, including exercises, lecture slides, corrections of mistakes, and other resources that should be useful to both readers and instructors.

vii

Acknowledgments This book would not have been possible without the contributions of many people. We would like to thank those who commented on our proposal for the book and helped plan its contents and organization: Guillaume Alain, Kyunghyun Cho, Çağlar Gülçehre, David Krueger, Hugo Larochelle, Razvan Pascanu and Thomas Rohée. We would like to thank the people who oﬀered feedback on the content of the book itself. Some oﬀered feedback on many chapters: Martín Abadi, Guillaume Alain, Ion Androutsopoulos, Fred Bertsch, Olexa Bilaniuk, Ufuk Can Biçici, Matko Bošnjak, John Boersma, Greg Brockman, Alexandre de Brébisson, Pierre Luc Carrier, Sarath Chandar, Pawel Chilinski, Mark Daoust, Oleg Dashevskii, Laurent Dinh, Stephan Dreseitl, Jim Fan, Miao Fan, Meire Fortunato, Frédéric Francis, Nando de Freitas, Çağlar Gülçehre, Jurgen Van Gael, Javier Alonso García, Jonathan Hunt, Gopi Jeyaram, Chingiz Kabytayev, Lukasz Kaiser, Varun Kanade, Akiel Khan, John King, Diederik P. Kingma, Yann LeCun, Rudolf Mathey, Matías Mattamala, Abhinav Maurya, Kevin Murphy, Oleg Mürk, Roman Novak, Augustus Q. Odena, Simon Pavlik, Karl Pichotta, Kari Pulli, Roussel Rahman, Tapani Raiko, Anurag Ranjan, Johannes Roith, Mihaela Rosca, Halis Sak, César Salgado, Grigory Sapunov, Yoshinori Sasaki, Mike Schuster, Julian Serban, Nir Shabat, Ken Shirriﬀ, Andre Simpelo, Scott Stanley, David Sussillo, Ilya Sutskever, Carles Gelada Sáez, Graham Taylor, Valentin Tolmer, An Tran, Shubhendu Trivedi, Alexey Umnov, Vincent Vanhoucke, Marco Visentini-Scarzanella, David Warde-Farley, Dustin Webb, Kelvin Xu, Wei Xue, Ke Yang, Li Yao, Zygmunt Zając and Ozan Çağlayan. We would also like to thank those who provided us with useful feedback on individual chapters: • Notation: Zhang Yuanhang. • Chapter 1, Introduction: Yusuf Akgul, Sebastien Bratieres, Samira Ebrahimi, Charlie Gorichanaz, Brendan Loudermilk, Eric Morris, Cosmin Pârvulescu viii

CONTENTS

and Alfredo Solano. • Chapter 2, Linear Algebra: Amjad Almahairi, Nikola Banić, Kevin Bennett, Philippe Castonguay, Oscar Chang, Eric Fosler-Lussier, Andrey Khalyavin, Sergey Oreshkov, István Petrás, Dennis Prangle, Thomas Rohée, Colby Toland, Massimiliano Tomassoli, Alessandro Vitale and Bob Welland. • Chapter 3, Probability and Information Theory: John Philip Anderson, Kai Arulkumaran, Vincent Dumoulin, Rui Fa, Stephan Gouws, Artem Oboturov, Antti Rasmus, Alexey Surkov and Volker Tresp. • Chapter 4, Numerical Computation: Tran Lam An, Ian Fischer, and Hu Yuhuang. • Chapter 5, Machine Learning Basics: Dzmitry Bahdanau, Nikhil Garg, Makoto Otsuka, Bob Pepin, Philip Popien, Emmanuel Rayner, Kee-Bong Song, Zheng Sun and Andy Wu. • Chapter 6, Deep Feedforward Networks: Uriel Berdugo, Fabrizio Bottarel, Elizabeth Burl, Ishan Durugkar, Jeﬀ Hlywa, Jong Wook Kim, David Krueger and Aditya Kumar Praharaj. • Chapter 7, Regularization for Deep Learning: Kshitij Lauria, Inkyu Lee, Sunil Mohan and Joshua Salisbury. • Chapter 8, Optimization for Training Deep Models: Marcel Ackermann, Rowel Atienza, Andrew Brock, Tegan Maharaj, James Martens, Klaus Strobl and Martin Vita. • Chapter 9, Convolutional Networks: Martín Arjovsky, Eugene Brevdo, Konstantin Divilov, Eric Jensen, Asifullah Khan, Mehdi Mirza, Alex Paino, Eddie Pierce, Marjorie Sayer, Ryan Stout and Wentao Wu. • Chapter 10, Sequence Modeling: Recurrent and Recursive Nets: Gökçen Eraslan, Steven Hickson, Razvan Pascanu, Lorenzo von Ritter, Rui Rodrigues, Dmitriy Serdyuk, Dongyu Shi and Kaiyu Yang. • Chapter 11, Practical Methodology: Daniel Beckstein. • Chapter 12, Applications: George Dahl and Ribana Roscher. • Chapter 15, Representation Learning: Kunal Ghosh. ix

CONTENTS

• Chapter 16, Structured Probabilistic Models for Deep Learning: Minh Lê and Anton Varfolom. • Chapter 18, Confronting the Partition Function: Sam Bowman. • Chapter 19, Approximate Inference: Yujia Bao. • Chapter 20, Deep Generative Models: Nicolas Chapados, Daniel Galvez, Wenming Ma, Fady Medhat, Shakir Mohamed and Grégoire Montavon. • Bibliography: Lukas Michelbacher and Leslie N. Smith. We also want to thank those who allowed us to reproduce images, ﬁgures or data from their publications. We indicate their contributions in the ﬁgure captions throughout the text. We would like to thank Lu Wang for writing pdf2htmlEX, which we used to make the web version of the book, and for oﬀering support to improve the quality of the resulting HTML. We would like to thank Ian’s wife Daniela Flori Goodfellow for patiently supporting Ian during the writing of the book as well as for help with proofreading. We would like to thank the Google Brain team for providing an intellectual environment where Ian could devote a tremendous amount of time to writing this book and receive feedback and guidance from colleagues. We would especially like to thank Ian’s former manager, Greg Corrado, and his current manager, Samy Bengio, for their support of this project. Finally, we would like to thank Geoﬀrey Hinton for encouragement when writing was diﬃcult.

x

Notation This section provides a concise reference describing the notation used throughout this book. If you are unfamiliar with any of the corresponding mathematical concepts, this notation reference may seem intimidating. However, do not despair, we describe most of these ideas in chapters 2-4. Numbers and Arrays a

A scalar (integer or real)

a

A vector

A

A matrix

A

A tensor

In

Identity matrix with n rows and n columns

I

Identity matrix with dimensionality implied by context

e(i)

Standard basis vector [0, . . . , 0, 1, 0, . . . , 0] with a 1 at position i

diag(a)

A square, diagonal matrix with diagonal entries given by a

a

A scalar random variable

a

A vector-valued random variable

A

A matrix-valued random variable

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CONTENTS

Sets and Graphs A

A set

R

The set of real numbers

{0, 1}

The set containing 0 and 1

{0, 1, . . . , n}

The set of all integers between 0 and n

[a, b]

The real interval including a and b

(a, b]

The real interval excluding a but including b

A\B

Set subtraction, i.e., the set containing the elements of A that are not in B

G

A graph

P a G(x i)

The parents of xi in G Indexing

ai

Element i of vector a, with indexing starting at 1

a−i

All elements of vector a except for element i

Ai,j

Element i, j of matrix A

Ai,:

Row i of matrix A

A:,i

Column i of matrix A

Ai,j,k

Element (i, j, k) of a 3-D tensor A

A:,:,i

2-D slice of a 3-D tensor

ai

Element i of the random vector a Linear Algebra Operations

A

Transpose of matrix A

A+

Moore-Penrose pseudoinverse of A

AB

Element-wise (Hadamard) product of A and B

det(A)

Determinant of A

xii

CONTENTS

Calculus Derivative of y with respect to x

dy dx ∂y ∂x ∇x y

Partial derivative of y with respect to x Gradient of y with respect to x

∇X y

Matrix derivatives of y with respect to X

∇Xy

Tensor containing derivatives of y with respect to X

∂f ∂x 2 ∇x f (x) or H (f )(x) f (x)dx f (x)dx

Jacobian matrix J ∈ Rm×n of f : Rn → Rm The Hessian matrix of f at input point x Deﬁnite integral over the entire domain of x Deﬁnite integral with respect to x over the set S

S

Probability and Information Theory a ⊥b

The random variables a and b are independent

a ⊥b | c

They are are conditionally independent given c

P (a )

A probability distribution over a discrete variable

p(a)

A probability distribution over a continuous variable, or over a variable whose type has not been speciﬁed

a∼P Ex∼P [f (x)] or Ef (x) Var(f (x)) Cov(f (x), g(x)) H(x)

Random variable a has distribution P Expectation of f (x) with respect to P (x) Variance of f (x) under P (x) Covariance of f (x) and g(x) under P (x) Shannon entropy of the random variable x

DKL(P Q)

Kullback-Leibler divergence of P and Q

N (x; µ, Σ)

Gaussian distribution over x with mean µ and covariance Σ

xiii

CONTENTS

f :A→B f ◦g f (x; θ)

Functions The function f with domain A and range B Composition of the functions f and g A function of x parametrized by θ. Sometimes we just write f (x) and ignore the argument θ to lighten notation.

log x

Natural logarithm of x

σ (x )

Logistic sigmoid,

ζ (x )

Softplus, log(1 + exp(x))

||x||p

Lp norm of x

||x|| x+

1 1 + exp(−x)

L2 norm of x Positive part of x, i.e., max(0, x)

1 condition is 1 if the condition is true, 0 otherwise Sometimes we use a function f whose argument is a scalar, but apply it to a vector, matrix, or tensor: f (x), f (X), or f(X). This means to apply f to the array element-wise. For example, if C = σ (X), then C i,j,k = σ(Xi,j,k ) for all valid values of i, j and k.

p data pˆdata X x(i)

Datasets and distributions The data generating distribution The empirical distribution deﬁned by the training set A set of training examples The i-th example (input) from a dataset

y (i) or y(i)

The target associated with x(i) for supervised learning

X

The m × n matrix with input example x(i) in row Xi,:

xiv

Chapter 1

Introduction Inventors have long dreamed of creating machines that think. This desire dates back to at least the time of ancient Greece. The mythical ﬁgures Pygmalion, Daedalus, and Hephaestus may all be interpreted as legendary inventors, and Galatea, Talos, and Pandora may all be regarded as artiﬁcial life (Ovid and Martin, 2004; Sparkes, 1996; Tandy, 1997). When programmable computers were ﬁrst conceived, people wondered whether they might become intelligent, over a hundred years before one was built (Lovelace, 1842). Today, artiﬁcial intelligence (AI) is a thriving ﬁeld with many practical applications and active research topics. We look to intelligent software to automate routine labor, understand speech or images, make diagnoses in medicine and support basic scientiﬁc research. In the early days of artiﬁcial intelligence, the ﬁeld rapidly tackled and solved problems that are intellectually diﬃcult for human beings but relatively straightforward for computers—problems that can be described by a list of formal, mathematical rules. The true challenge to artiﬁcial intelligence proved to be solving the tasks that are easy for people to perform but hard for people to describe formally—problems that we solve intuitively, that feel automatic, like recognizing spoken words or faces in images. This book is about a solution to these more intuitive problems. This solution is to allow computers to learn from experience and understand the world in terms of a hierarchy of concepts, with each concept deﬁned in terms of its relation to simpler concepts. By gathering knowledge from experience, this approach avoids the need for human operators to formally specify all of the knowledge that the computer needs. The hierarchy of concepts allows the computer to learn complicated concepts by building them out of simpler ones. If we draw a graph showing how these 1

CHAPTER 1. INTRODUCTION

concepts are built on top of each other, the graph is deep, with many layers. For this reason, we call this approach to AI deep learning. Many of the early successes of AI took place in relatively sterile and formal environments and did not require computers to have much knowledge about the world. For example, IBM’s Deep Blue chess-playing system defeated world champion Garry Kasparov in 1997 (Hsu, 2002). Chess is of course a very simple world, containing only sixty-four locations and thirty-two pieces that can move in only rigidly circumscribed ways. Devising a successful chess strategy is a tremendous accomplishment, but the challenge is not due to the diﬃculty of describing the set of chess pieces and allowable moves to the computer. Chess can be completely described by a very brief list of completely formal rules, easily provided ahead of time by the programmer. Ironically, abstract and formal tasks that are among the most diﬃcult mental undertakings for a human being are among the easiest for a computer. Computers have long been able to defeat even the best human chess player, but are only recently matching some of the abilities of average human beings to recognize objects or speech. A person’s everyday life requires an immense amount of knowledge about the world. Much of this knowledge is subjective and intuitive, and therefore diﬃcult to articulate in a formal way. Computers need to capture this same knowledge in order to behave in an intelligent way. One of the key challenges in artiﬁcial intelligence is how to get this informal knowledge into a computer. Several artiﬁcial intelligence projects have sought to hard-code knowledge about the world in formal languages. A computer can reason about statements in these formal languages automatically using logical inference rules. This is known as the knowledge base approach to artiﬁcial intelligence. None of these projects has led to a major success. One of the most famous such projects is Cyc (Lenat and Guha, 1989). Cyc is an inference engine and a database of statements in a language called CycL. These statements are entered by a staﬀ of human supervisors. It is an unwieldy process. People struggle to devise formal rules with enough complexity to accurately describe the world. For example, Cyc failed to understand a story about a person named Fred shaving in the morning (Linde, 1992). Its inference engine detected an inconsistency in the story: it knew that people do not have electrical parts, but because Fred was holding an electric razor, it believed the entity “FredWhileShaving” contained electrical parts. It therefore asked whether Fred was still a person while he was shaving. The diﬃculties faced by systems relying on hard-coded knowledge suggest that AI systems need the ability to acquire their own knowledge, by extracting patterns from raw data. This capability is known as machine learning. The introduction 2

CHAPTER 1. INTRODUCTION

of machine learning allowed computers to tackle problems involving knowledge of the real world and make decisions that appear subjective. A simple machine learning algorithm called logistic regression can determine whether to recommend cesarean delivery (Mor-Yosef et al., 1990). A simple machine learning algorithm called naive Bayes can separate legitimate e-mail from spam e-mail. The performance of these simple machine learning algorithms depends heavily on the representation of the data they are given. For example, when logistic regression is used to recommend cesarean delivery, the AI system does not examine the patient directly. Instead, the doctor tells the system several pieces of relevant information, such as the presence or absence of a uterine scar. Each piece of information included in the representation of the patient is known as a feature. Logistic regression learns how each of these features of the patient correlates with various outcomes. However, it cannot inﬂuence the way that the features are deﬁned in any way. If logistic regression was given an MRI scan of the patient, rather than the doctor’s formalized report, it would not be able to make useful predictions. Individual pixels in an MRI scan have negligible correlation with any complications that might occur during delivery. This dependence on representations is a general phenomenon that appears throughout computer science and even daily life. In computer science, operations such as searching a collection of data can proceed exponentially faster if the collection is structured and indexed intelligently. People can easily perform arithmetic on Arabic numerals, but ﬁnd arithmetic on Roman numerals much more time-consuming. It is not surprising that the choice of representation has an enormous eﬀect on the performance of machine learning algorithms. For a simple visual example, see Fig. 1.1. Many artiﬁcial intelligence tasks can be solved by designing the right set of features to extract for that task, then providing these features to a simple machine learning algorithm. For example, a useful feature for speaker identiﬁcation from sound is an estimate of the size of speaker’s vocal tract. It therefore gives a strong clue as to whether the speaker is a man, woman, or child. However, for many tasks, it is diﬃcult to know what features should be extracted. For example, suppose that we would like to write a program to detect cars in photographs. We know that cars have wheels, so we might like to use the presence of a wheel as a feature. Unfortunately, it is diﬃcult to describe exactly what a wheel looks like in terms of pixel values. A wheel has a simple geometric shape but its image may be complicated by shadows falling on the wheel, the sun glaring oﬀ the metal parts of the wheel, the fender of the car or an object in the foreground obscuring part of the wheel, and so on. 3

CHAPTER 1. INTRODUCTION

Figure 1.1: Example of diﬀerent representations: suppose we want to separate two categories of data by drawing a line between them in a scatterplot. In the plot on the left, we represent some data using Cartesian coordinates, and the task is impossible. In the plot on the right, we represent the data with polar coordinates and the task becomes simple to solve with a vertical line. (Figure produced in collaboration with David Warde-Farley)

One solution to this problem is to use machine learning to discover not only the mapping from representation to output but also the representation itself. This approach is known as representation learning. Learned representations often result in much better performance than can be obtained with hand-designed representations. They also allow AI systems to rapidly adapt to new tasks, with minimal human intervention. A representation learning algorithm can discover a good set of features for a simple task in minutes, or a complex task in hours to months. Manually designing features for a complex task requires a great deal of human time and eﬀort; it can take decades for an entire community of researchers. The quintessential example of a representation learning algorithm is the autoencoder. An autoencoder is the combination of an encoder function that converts the input data into a diﬀerent representation, and a decoder function that converts the new representation back into the original format. Autoencoders are trained to preserve as much information as possible when an input is run through the encoder and then the decoder, but are also trained to make the new representation have various nice properties. Diﬀerent kinds of autoencoders aim to achieve diﬀerent kinds of properties. When designing features or algorithms for learning features, our goal is usually to separate the factors of variation that explain the observed data. In this context, we use the word “factors” simply to refer to separate sources of inﬂuence; the factors are usually not combined by multiplication. Such factors are often not quantities 4

CHAPTER 1. INTRODUCTION

that are directly observed. Instead, they may exist either as unobserved objects or unobserved forces in the physical world that aﬀect observable quantities. They may also exist as constructs in the human mind that provide useful simplifying explanations or inferred causes of the observed data. They can be thought of as concepts or abstractions that help us make sense of the rich variability in the data. When analyzing a speech recording, the factors of variation include the speaker’s age, their sex, their accent and the words that they are speaking. When analyzing an image of a car, the factors of variation include the position of the car, its color, and the angle and brightness of the sun. A major source of diﬃculty in many real-world artiﬁcial intelligence applications is that many of the factors of variation inﬂuence every single piece of data we are able to observe. The individual pixels in an image of a red car might be very close to black at night. The shape of the car’s silhouette depends on the viewing angle. Most applications require us to disentangle the factors of variation and discard the ones that we do not care about. Of course, it can be very diﬃcult to extract such high-level, abstract features from raw data. Many of these factors of variation, such as a speaker’s accent, can be identiﬁed only using sophisticated, nearly human-level understanding of the data. When it is nearly as diﬃcult to obtain a representation as to solve the original problem, representation learning does not, at ﬁrst glance, seem to help us. Deep learning solves this central problem in representation learning by introducing representations that are expressed in terms of other, simpler representations. Deep learning allows the computer to build complex concepts out of simpler concepts. Fig. 1.2 shows how a deep learning system can represent the concept of an image of a person by combining simpler concepts, such as corners and contours, which are in turn deﬁned in terms of edges. The quintessential example of a deep learning model is the feedforward deep network or multilayer perceptron (MLP). A multilayer perceptron is just a mathematical function mapping some set of input values to output values. The function is formed by composing many simpler functions. We can think of each application of a diﬀerent mathematical function as providing a new representation of the input. The idea of learning the right representation for the data provides one perspective on deep learning. Another perspective on deep learning is that depth allows the computer to learn a multi-step computer program. Each layer of the representation can be thought of as the state of the computer’s memory after executing another set of instructions in parallel. Networks with greater depth can execute more instructions in sequence. Sequential instructions oﬀer great power because later instructions can refer back to the results of earlier instructions. According to this 5

CHAPTER 1. INTRODUCTION

CAR

PERSON

ANIMAL

Output (object identity)

3rd hidden layer (object parts)

2nd hidden layer (corners and contours)

1st hidden layer (edges)

Visible layer (input pixels)

Figure 1.2: Illustration of a deep learning model. It is diﬃcult for a computer to understand the meaning of raw sensory input data, such as this image represented as a collection of pixel values. The function mapping from a set of pixels to an object identity is very complicated. Learning or evaluating this mapping seems insurmountable if tackled directly. Deep learning resolves this diﬃculty by breaking the desired complicated mapping into a series of nested simple mappings, each described by a diﬀerent layer of the model. The input is presented at the visible layer, so named because it contains the variables that we are able to observe. Then a series of hidden layers extracts increasingly abstract features from the image. These layers are called “hidden” because their values are not given in the data; instead the model must determine which concepts are useful for explaining the relationships in the observed data. The images here are visualizations of the kind of feature represented by each hidden unit. Given the pixels, the ﬁrst layer can easily identify edges, by comparing the brightness of neighboring pixels. Given the ﬁrst hidden layer’s description of the edges, the second hidden layer can easily search for corners and extended contours, which are recognizable as collections of edges. Given the second hidden layer’s description of the image in terms of corners and contours, the third hidden layer can detect entire parts of speciﬁc objects, by ﬁnding speciﬁc collections of contours and corners. Finally, this description of the image in terms of the object parts it contains can be used to recognize the objects present in the image. Images reproduced with permission from Zeiler and Fergus (2014).

6

CHAPTER 1. INTRODUCTION

σ

Element Set

+ × σ

+

× w1

Element Set

x1

× w2

Logistic Regression

x2

Logistic Regression

w

x

Figure 1.3: Illustration of computational graphs mapping an input to an output where each node performs an operation. Depth is the length of the longest path from input to output but depends on the deﬁnition of what constitutes a possible computational step. The computation depicted in these graphs is the output of a logistic regression model, σ(wT x), where σ is the logistic sigmoid function. If we use addition, multiplication and logistic sigmoids as the elements of our computer language, then this model has depth three. If we view logistic regression as an element itself, then this model has depth one.

view of deep learning, not all of the information in a layer’s activations necessarily encodes factors of variation that explain the input. The representation also stores state information that helps to execute a program that can make sense of the input. This state information could be analogous to a counter or pointer in a traditional computer program. It has nothing to do with the content of the input speciﬁcally, but it helps the model to organize its processing. There are two main ways of measuring the depth of a model. The ﬁrst view is based on the number of sequential instructions that must be executed to evaluate the architecture. We can think of this as the length of the longest path through a ﬂow chart that describes how to compute each of the model’s outputs given its inputs. Just as two equivalent computer programs will have diﬀerent lengths depending on which language the program is written in, the same function may be drawn as a ﬂowchart with diﬀerent depths depending on which functions we allow to be used as individual steps in the ﬂowchart. Fig. 1.3 illustrates how this choice of language can give two diﬀerent measurements for the same architecture. Another approach, used by deep probabilistic models, regards the depth of a model as being not the depth of the computational graph but the depth of the graph describing how concepts are related to each other. In this case, the depth of the ﬂowchart of the computations needed to compute the representation of 7

CHAPTER 1. INTRODUCTION

each concept may be much deeper than the graph of the concepts themselves. This is because the system’s understanding of the simpler concepts can be reﬁned given information about the more complex concepts. For example, an AI system observing an image of a face with one eye in shadow may initially only see one eye. After detecting that a face is present, it can then infer that a second eye is probably present as well. In this case, the graph of concepts only includes two layers—a layer for eyes and a layer for faces—but the graph of computations includes 2n layers if we reﬁne our estimate of each concept given the other n times. Because it is not always clear which of these two views—the depth of the computational graph, or the depth of the probabilistic modeling graph—is most relevant, and because diﬀerent people choose diﬀerent sets of smallest elements from which to construct their graphs, there is no single correct value for the depth of an architecture, just as there is no single correct value for the length of a computer program. Nor is there a consensus about how much depth a model requires to qualify as “deep.” However, deep learning can safely be regarded as the study of models that either involve a greater amount of composition of learned functions or learned concepts than traditional machine learning does. To summarize, deep learning, the subject of this book, is an approach to AI. Speciﬁcally, it is a type of machine learning, a technique that allows computer systems to improve with experience and data. According to the authors of this book, machine learning is the only viable approach to building AI systems that can operate in complicated, real-world environments. Deep learning is a particular kind of machine learning that achieves great power and ﬂexibility by learning to represent the world as a nested hierarchy of concepts, with each concept deﬁned in relation to simpler concepts, and more abstract representations computed in terms of less abstract ones. Fig. 1.4 illustrates the relationship between these diﬀerent AI disciplines. Fig. 1.5 gives a high-level schematic of how each works.

1.1

Who Should Read This Book?

This book can be useful for a variety of readers, but we wrote it with two main target audiences in mind. One of these target audiences is university students (undergraduate or graduate) learning about machine learning, including those who are beginning a career in deep learning and artiﬁcial intelligence research. The other target audience is software engineers who do not have a machine learning or statistics background, but want to rapidly acquire one and begin using deep learning in their product or platform. Deep learning has already proven useful in many software disciplines including computer vision, speech and audio processing, 8

CHAPTER 1. INTRODUCTION

Deep learning Example: MLPs

Example: Shallow autoencoders

Example: Logistic regression

Example: Knowledge bases

Representation learning

Machine learning

AI

Figure 1.4: A Venn diagram showing how deep learning is a kind of representation learning, which is in turn a kind of machine learning, which is used for many but not all approaches to AI. Each section of the Venn diagram includes an example of an AI technology.

9

CHAPTER 1. INTRODUCTION

Output

Output

Output

Mapping from features

Output

Mapping from features

Mapping from features

Additional layers of more abstract features

Handdesigned program

Handdesigned features

Features

Simple features

Input

Input

Input

Input

Rule-based systems

Deep learning

Classic machine learning

Representation learning

Figure 1.5: Flowcharts showing how the diﬀerent parts of an AI system relate to each other within diﬀerent AI disciplines. Shaded boxes indicate components that are able to learn from data.

10

CHAPTER 1. INTRODUCTION

natural language processing, robotics, bioinformatics and chemistry, video games, search engines, online advertising and ﬁnance. This book has been organized into three parts in order to best accommodate a variety of readers. Part I introduces basic mathematical tools and machine learning concepts. Part II describes the most established deep learning algorithms that are essentially solved technologies. Part III describes more speculative ideas that are widely believed to be important for future research in deep learning. Readers should feel free to skip parts that are not relevant given their interests or background. Readers familiar with linear algebra, probability, and fundamental machine learning concepts can skip Part I, for example, while readers who just want to implement a working system need not read beyond Part II. To help choose which chapters to read, Fig. 1.6 provides a ﬂowchart showing the high-level organization of the book. We do assume that all readers come from a computer science background. We assume familiarity with programming, a basic understanding of computational performance issues, complexity theory, introductory level calculus and some of the terminology of graph theory.

1.2

Historical Trends in Deep Learning

It is easiest to understand deep learning with some historical context. Rather than providing a detailed history of deep learning, we identify a few key trends: • Deep learning has had a long and rich history, but has gone by many names reﬂecting diﬀerent philosophical viewpoints, and has waxed and waned in popularity. • Deep learning has become more useful as the amount of available training data has increased. • Deep learning models have grown in size over time as computer hardware and software infrastructure for deep learning has improved. • Deep learning has solved increasingly complicated applications with increasing accuracy over time.

11

CHAPTER 1. INTRODUCTION

1. Introduction

Part I: Applied Math and Machine Learning Basics 2. Linear Algebra

3. Probability and Information Theory

4. Numerical Computation

5. Machine Learning Basics

Part II: Deep Networks: Modern Practices 6. Deep Feedforward Networks

7. Regularization

8. Optimization

11. Practical Methodology

9. CNNs

10. RNNs

12. Applications

Part III: Deep Learning Research 13. Linear Factor Models

14. Autoencoders

15. Representation Learning

16. Structured Probabilistic Models

17. Monte Carlo Methods

19. Inference

18. Partition Function

20. Deep Generative Models

Figure 1.6: The high-level organization of the book. An arrow from one chapter to another indicates that the former chapter is prerequisite material for understanding the latter. 12

CHAPTER 1. INTRODUCTION

1.2.1

The Many Names and Changing Fortunes of Neural Networks

We expect that many readers of this book have heard of deep learning as an exciting new technology, and are surprised to see a mention of “history” in a book about an emerging ﬁeld. In fact, deep learning dates back to the 1940s. Deep learning only appears to be new, because it was relatively unpopular for several years preceding its current popularity, and because it has gone through many diﬀerent names, and has only recently become called “deep learning.” The ﬁeld has been rebranded many times, reﬂecting the inﬂuence of diﬀerent researchers and diﬀerent perspectives. A comprehensive history of deep learning is beyond the scope of this textbook. However, some basic context is useful for understanding deep learning. Broadly speaking, there have been three waves of development of deep learning: deep learning known as cybernetics in the 1940s–1960s, deep learning known as connectionism in the 1980s–1990s, and the current resurgence under the name deep learning beginning in 2006. This is quantitatively illustrated in Fig. 1.7. Some of the earliest learning algorithms we recognize today were intended to be computational models of biological learning, i.e. models of how learning happens or could happen in the brain. As a result, one of the names that deep learning has gone by is artiﬁcial neural networks (ANNs). The corresponding perspective on deep learning models is that they are engineered systems inspired by the biological brain (whether the human brain or the brain of another animal). While the kinds of neural networks used for machine learning have sometimes been used to understand brain function (Hinton and Shallice, 1991), they are generally not designed to be realistic models of biological function. The neural perspective on deep learning is motivated by two main ideas. One idea is that the brain provides a proof by example that intelligent behavior is possible, and a conceptually straightforward path to building intelligence is to reverse engineer the computational principles behind the brain and duplicate its functionality. Another perspective is that it would be deeply interesting to understand the brain and the principles that underlie human intelligence, so machine learning models that shed light on these basic scientiﬁc questions are useful apart from their ability to solve engineering applications. The modern term “deep learning” goes beyond the neuroscientiﬁc perspective on the current breed of machine learning models. It appeals to a more general principle of learning multiple levels of composition, which can be applied in machine learning frameworks that are not necessarily neurally inspired. 13

Frequency of Word or Phrase

CHAPTER 1. INTRODUCTION

0.000250 0.000200

cybernetics (connectionism + neural networks)

0.000150 0.000100 0.000050 0.000000 1940

1950

1960

1970

1980

1990

2000

Year

Figure 1.7: The ﬁgure shows two of the three historical waves of artiﬁcial neural nets research, as measured by the frequency of the phrases “cybernetics” and “connectionism” or “neural networks” according to Google Books (the third wave is too recent to appear). The ﬁrst wave started with cybernetics in the 1940s–1960s, with the development of theories of biological learning (McCulloch and Pitts, 1943; Hebb, 1949) and implementations of the ﬁrst models such as the perceptron (Rosenblatt, 1958) allowing the training of a single neuron. The second wave started with the connectionist approach of the 1980–1995 period, with back-propagation (Rumelhart et al., 1986a) to train a neural network with one or two hidden layers. The current and third wave, deep learning, started around 2006 (Hinton et al., 2006; Bengio et al., 2007; Ranzato et al., 2007a), and is just now appearing in book form as of 2016. The other two waves similarly appeared in book form much later than the corresponding scientiﬁc activity occurred.

14

CHAPTER 1. INTRODUCTION

The earliest predecessors of modern deep learning were simple linear models motivated from a neuroscientiﬁc perspective. These models were designed to take a set of n input values x1, . . . , xn and associate them with an output y. These models would learn a set of weights w1, . . . , wn and compute their output f(x, w ) = x 1w1 + · · · + xn wn . This ﬁrst wave of neural networks research was known as cybernetics, as illustrated in Fig. 1.7. The McCulloch-Pitts Neuron (McCulloch and Pitts, 1943) was an early model of brain function. This linear model could recognize two diﬀerent categories of inputs by testing whether f (x, w ) is positive or negative. Of course, for the model to correspond to the desired deﬁnition of the categories, the weights needed to be set correctly. These weights could be set by the human operator. In the 1950s, the perceptron (Rosenblatt, 1958, 1962) became the ﬁrst model that could learn the weights deﬁning the categories given examples of inputs from each category. The adaptive linear element (ADALINE), which dates from about the same time, simply returned the value of f (x) itself to predict a real number (Widrow and Hoﬀ, 1960), and could also learn to predict these numbers from data. These simple learning algorithms greatly aﬀected the modern landscape of machine learning. The training algorithm used to adapt the weights of the ADALINE was a special case of an algorithm called stochastic gradient descent. Slightly modiﬁed versions of the stochastic gradient descent algorithm remain the dominant training algorithms for deep learning models today. Models based on the f(x, w ) used by the perceptron and ADALINE are called linear models. These models remain some of the most widely used machine learning models, though in many cases they are trained in diﬀerent ways than the original models were trained. Linear models have many limitations. Most famously, they cannot learn the XOR function, where f ([0,1] , w) = 1 and f ([1, 0], w) = 1 but f ([1, 1], w) = 0 and f ([0, 0], w) = 0. Critics who observed these ﬂaws in linear models caused a backlash against biologically inspired learning in general (Minsky and Papert, 1969). This was the ﬁrst major dip in the popularity of neural networks. Today, neuroscience is regarded as an important source of inspiration for deep learning researchers, but it is no longer the predominant guide for the ﬁeld. The main reason for the diminished role of neuroscience in deep learning research today is that we simply do not have enough information about the brain to use it as a guide. To obtain a deep understanding of the actual algorithms used by the brain, we would need to be able to monitor the activity of (at the very least) thousands of interconnected neurons simultaneously. Because we are not able to do this, we are far from understanding even some of the most simple and 15

CHAPTER 1. INTRODUCTION

well-studied parts of the brain (Olshausen and Field, 2005). Neuroscience has given us a reason to hope that a single deep learning algorithm can solve many diﬀerent tasks. Neuroscientists have found that ferrets can learn to “see” with the auditory processing region of their brain if their brains are rewired to send visual signals to that area (Von Melchner et al., 2000). This suggests that much of the mammalian brain might use a single algorithm to solve most of the diﬀerent tasks that the brain solves. Before this hypothesis, machine learning research was more fragmented, with diﬀerent communities of researchers studying natural language processing, vision, motion planning and speech recognition. Today, these application communities are still separate, but it is common for deep learning research groups to study many or even all of these application areas simultaneously. We are able to draw some rough guidelines from neuroscience. The basic idea of having many computational units that become intelligent only via their interactions with each other is inspired by the brain. The Neocognitron (Fukushima, 1980) introduced a powerful model architecture for processing images that was inspired by the structure of the mammalian visual system and later became the basis for the modern convolutional network (LeCun et al., 1998b), as we will see in Sec. 9.10. Most neural networks today are based on a model neuron called the rectiﬁed linear unit. The original Cognitron (Fukushima, 1975) introduced a more complicated version that was highly inspired by our knowledge of brain function. The simpliﬁed modern version was developed incorporating ideas from many viewpoints, with Nair and Hinton (2010) and Glorot et al. (2011a) citing neuroscience as an inﬂuence, and Jarrett et al. (2009) citing more engineering-oriented inﬂuences. While neuroscience is an important source of inspiration, it need not be taken as a rigid guide. We know that actual neurons compute very diﬀerent functions than modern rectiﬁed linear units, but greater neural realism has not yet led to an improvement in machine learning performance. Also, while neuroscience has successfully inspired several neural network architectures, we do not yet know enough about biological learning for neuroscience to oﬀer much guidance for the learning algorithms we use to train these architectures. Media accounts often emphasize the similarity of deep learning to the brain. While it is true that deep learning researchers are more likely to cite the brain as an inﬂuence than researchers working in other machine learning ﬁelds such as kernel machines or Bayesian statistics, one should not view deep learning as an attempt to simulate the brain. Modern deep learning draws inspiration from many ﬁelds, especially applied math fundamentals like linear algebra, probability, information theory, and numerical optimization. While some deep learning researchers cite neuroscience as an important source of inspiration, others are not concerned with 16

CHAPTER 1. INTRODUCTION

neuroscience at all. It is worth noting that the eﬀort to understand how the brain works on an algorithmic level is alive and well. This endeavor is primarily known as “computational neuroscience” and is a separate ﬁeld of study from deep learning. It is common for researchers to move back and forth between both ﬁelds. The ﬁeld of deep learning is primarily concerned with how to build computer systems that are able to successfully solve tasks requiring intelligence, while the ﬁeld of computational neuroscience is primarily concerned with building more accurate models of how the brain actually works. In the 1980s, the second wave of neural network research emerged in great part via a movement called connectionism or parallel distributed processing (Rumelhart et al., 1986c; McClelland et al., 1995). Connectionism arose in the context of cognitive science. Cognitive science is an interdisciplinary approach to understanding the mind, combining multiple diﬀerent levels of analysis. During the early 1980s, most cognitive scientists studied models of symbolic reasoning. Despite their popularity, symbolic models were diﬃcult to explain in terms of how the brain could actually implement them using neurons. The connectionists began to study models of cognition that could actually be grounded in neural implementations (Touretzky and Minton, 1985), reviving many ideas dating back to the work of psychologist Donald Hebb in the 1940s (Hebb, 1949). The central idea in connectionism is that a large number of simple computational units can achieve intelligent behavior when networked together. This insight applies equally to neurons in biological nervous systems and to hidden units in computational models. Several key concepts arose during the connectionism movement of the 1980s that remain central to today’s deep learning. One of these concepts is that of distributed representation (Hinton et al., 1986). This is the idea that each input to a system should be represented by many features, and each feature should be involved in the representation of many possible inputs. For example, suppose we have a vision system that can recognize cars, trucks, and birds and these objects can each be red, green, or blue. One way of representing these inputs would be to have a separate neuron or hidden unit that activates for each of the nine possible combinations: red truck, red car, red bird, green truck, and so on. This requires nine diﬀerent neurons, and each neuron must independently learn the concept of color and object identity. One way to improve on this situation is to use a distributed representation, with three neurons describing the color and three neurons describing the object identity. This requires only six neurons total instead of nine, and the neuron describing redness is able to learn about redness 17

CHAPTER 1. INTRODUCTION

from images of cars, trucks and birds, not only from images of one speciﬁc category of objects. The concept of distributed representation is central to this book, and will be described in greater detail in Chapter 15. Another major accomplishment of the connectionist movement was the successful use of back-propagation to train deep neural networks with internal representations and the popularization of the back-propagation algorithm (Rumelhart et al., 1986a; LeCun, 1987). This algorithm has waxed and waned in popularity but as of this writing is currently the dominant approach to training deep models. During the 1990s, researchers made important advances in modeling sequences with neural networks. Hochreiter (1991) and Bengio et al. (1994) identiﬁed some of the fundamental mathematical diﬃculties in modeling long sequences, described in Sec. 10.7. Hochreiter and Schmidhuber (1997) introduced the long short-term memory or LSTM network to resolve some of these diﬃculties. Today, the LSTM is widely used for many sequence modeling tasks, including many natural language processing tasks at Google. The second wave of neural networks research lasted until the mid-1990s. Ventures based on neural networks and other AI technologies began to make unrealistically ambitious claims while seeking investments. When AI research did not fulﬁll these unreasonable expectations, investors were disappointed. Simultaneously, other ﬁelds of machine learning made advances. Kernel machines (Boser et al., 1992; Cortes and Vapnik, 1995; Schölkopf et al., 1999) and graphical models (Jordan, 1998) both achieved good results on many important tasks. These two factors led to a decline in the popularity of neural networks that lasted until 2007. During this time, neural networks continued to obtain impressive performance on some tasks (LeCun et al., 1998b; Bengio et al., 2001). The Canadian Institute for Advanced Research (CIFAR) helped to keep neural networks research alive via its Neural Computation and Adaptive Perception (NCAP) research initiative. This program united machine learning research groups led by Geoﬀrey Hinton at University of Toronto, Yoshua Bengio at University of Montreal, and Yann LeCun at New York University. The CIFAR NCAP research initiative had a multi-disciplinary nature that also included neuroscientists and experts in human and computer vision. At this point in time, deep networks were generally believed to be very diﬃcult to train. We now know that algorithms that have existed since the 1980s work quite well, but this was not apparent circa 2006. The issue is perhaps simply that these algorithms were too computationally costly to allow much experimentation with the hardware available at the time. The third wave of neural networks research began with a breakthrough in 18

CHAPTER 1. INTRODUCTION

2006. Geoﬀrey Hinton showed that a kind of neural network called a deep belief network could be eﬃciently trained using a strategy called greedy layer-wise pretraining (Hinton et al., 2006), which will be described in more detail in Sec. 15.1. The other CIFAR-aﬃliated research groups quickly showed that the same strategy could be used to train many other kinds of deep networks (Bengio et al., 2007; Ranzato et al., 2007a) and systematically helped to improve generalization on test examples. This wave of neural networks research popularized the use of the term deep learning to emphasize that researchers were now able to train deeper neural networks than had been possible before, and to focus attention on the theoretical importance of depth (Bengio and LeCun, 2007; Delalleau and Bengio, 2011; Pascanu et al., 2014a; Montufar et al., 2014). At this time, deep neural networks outperformed competing AI systems based on other machine learning technologies as well as hand-designed functionality. This third wave of popularity of neural networks continues to the time of this writing, though the focus of deep learning research has changed dramatically within the time of this wave. The third wave began with a focus on new unsupervised learning techniques and the ability of deep models to generalize well from small datasets, but today there is more interest in much older supervised learning algorithms and the ability of deep models to leverage large labeled datasets.

1.2.2

Increasing Dataset Sizes

One may wonder why deep learning has only recently become recognized as a crucial technology though the ﬁrst experiments with artiﬁcial neural networks were conducted in the 1950s. Deep learning has been successfully used in commercial applications since the 1990s, but was often regarded as being more of an art than a technology and something that only an expert could use, until recently. It is true that some skill is required to get good performance from a deep learning algorithm. Fortunately, the amount of skill required reduces as the amount of training data increases. The learning algorithms reaching human performance on complex tasks today are nearly identical to the learning algorithms that struggled to solve toy problems in the 1980s, though the models we train with these algorithms have undergone changes that simplify the training of very deep architectures. The most important new development is that today we can provide these algorithms with the resources they need to succeed. Fig. 1.8 shows how the size of benchmark datasets has increased remarkably over time. This trend is driven by the increasing digitization of society. As more and more of our activities take place on computers, more and more of what we do is recorded. As our computers are increasingly networked together, it becomes easier to centralize these records and curate them 19

CHAPTER 1. INTRODUCTION

into a dataset appropriate for machine learning applications. The age of “Big Data” has made machine learning much easier because the key burden of statistical estimation—generalizing well to new data after observing only a small amount of data—has been considerably lightened. As of 2016, a rough rule of thumb is that a supervised deep learning algorithm will generally achieve acceptable performance with around 5,000 labeled examples per category, and will match or exceed human performance when trained with a dataset containing at least 10 million labeled examples. Working successfully with datasets smaller than this is an important research area, focusing in particular on how we can take advantage of large quantities of unlabeled examples, with unsupervised or semi-supervised learning.

1.2.3

Increasing Model Sizes

Another key reason that neural networks are wildly successful today after enjoying comparatively little success since the 1980s is that we have the computational resources to run much larger models today. One of the main insights of connectionism is that animals become intelligent when many of their neurons work together. An individual neuron or small collection of neurons is not particularly useful. Biological neurons are not especially densely connected. As seen in Fig. 1.10, our machine learning models have had a number of connections per neuron that was within an order of magnitude of even mammalian brains for decades. In terms of the total number of neurons, neural networks have been astonishingly small until quite recently, as shown in Fig. 1.11. Since the introduction of hidden units, artiﬁcial neural networks have doubled in size roughly every 2.4 years. This growth is driven by faster computers with larger memory and by the availability of larger datasets. Larger networks are able to achieve higher accuracy on more complex tasks. This trend looks set to continue for decades. Unless new technologies allow faster scaling, artiﬁcial neural networks will not have the same number of neurons as the human brain until at least the 2050s. Biological neurons may represent more complicated functions than current artiﬁcial neurons, so biological neural networks may be even larger than this plot portrays. In retrospect, it is not particularly surprising that neural networks with fewer neurons than a leech were unable to solve sophisticated artiﬁcial intelligence problems. Even today’s networks, which we consider quite large from a computational systems point of view, are smaller than the nervous system of even relatively primitive vertebrate animals like frogs. The increase in model size over time, due to the availability of faster CPUs, 20

CHAPTER 1. INTRODUCTION

Figure 1.8: Dataset sizes have increased greatly over time. In the early 1900s, statisticians studied datasets using hundreds or thousands of manually compiled measurements (Garson, 1900; Gosset, 1908; Anderson, 1935; Fisher, 1936). In the 1950s through 1980s, the pioneers of biologically inspired machine learning often worked with small, synthetic datasets, such as low-resolution bitmaps of letters, that were designed to incur low computational cost and demonstrate that neural networks were able to learn speciﬁc kinds of functions (Widrow and Hoﬀ, 1960; Rumelhart et al., 1986b). In the 1980s and 1990s, machine learning became more statistical in nature and began to leverage larger datasets containing tens of thousands of examples such as the MNIST dataset (shown in Fig. 1.9) of scans of handwritten numbers (LeCun et al., 1998b). In the ﬁrst decade of the 2000s, more sophisticated datasets of this same size, such as the CIFAR-10 dataset (Krizhevsky and Hinton, 2009) continued to be produced. Toward the end of that decade and throughout the ﬁrst half of the 2010s, signiﬁcantly larger datasets, containing hundreds of thousands to tens of millions of examples, completely changed what was possible with deep learning. These datasets included the public Street View House Numbers dataset (Netzer et al., 2011), various versions of the ImageNet dataset (Deng et al., 2009, 2010a; Russakovsky et al., 2014a), and the Sports-1M dataset (Karpathy et al., 2014). At the top of the graph, we see that datasets of translated sentences, such as IBM’s dataset constructed from the Canadian Hansard (Brown et al., 1990) and the WMT 2014 English to French dataset (Schwenk, 2014) are typically far ahead of other dataset sizes.

21

CHAPTER 1. INTRODUCTION

Figure 1.9: Example inputs from the MNIST dataset. The “NIST” stands for National Institute of Standards and Technology, the agency that originally collected this data. The “M” stands for “modiﬁed,” since the data has been preprocessed for easier use with machine learning algorithms. The MNIST dataset consists of scans of handwritten digits and associated labels describing which digit 0-9 is contained in each image. This simple classiﬁcation problem is one of the simplest and most widely used tests in deep learning research. It remains popular despite being quite easy for modern techniques to solve. Geoﬀrey Hinton has described it as “the drosophila of machine learning,” meaning that it allows machine learning researchers to study their algorithms in controlled laboratory conditions, much as biologists often study fruit ﬂies.

22

CHAPTER 1. INTRODUCTION

the advent of general purpose GPUs (described in Sec. 12.1.2), faster network connectivity and better software infrastructure for distributed computing, is one of the most important trends in the history of deep learning. This trend is generally expected to continue well into the future.

1.2.4

Increasing Accuracy, Complexity and Real-World Impact

Since the 1980s, deep learning has consistently improved in its ability to provide accurate recognition or prediction. Moreover, deep learning has consistently been applied with success to broader and broader sets of applications. The earliest deep models were used to recognize individual objects in tightly cropped, extremely small images (Rumelhart et al., 1986a). Since then there has been a gradual increase in the size of images neural networks could process. Modern object recognition networks process rich high-resolution photographs and do not have a requirement that the photo be cropped near the object to be recognized (Krizhevsky et al., 2012). Similarly, the earliest networks could only recognize two kinds of objects (or in some cases, the absence or presence of a single kind of object), while these modern networks typically recognize at least 1,000 diﬀerent categories of objects. The largest contest in object recognition is the ImageNet Large-Scale Visual Recognition Challenge (ILSVRC) held each year. A dramatic moment in the meteoric rise of deep learning came when a convolutional network won this challenge for the ﬁrst time and by a wide margin, bringing down the state-of-the-art top-5 error rate from 26.1% to 15.3% (Krizhevsky et al., 2012), meaning that the convolutional network produces a ranked list of possible categories for each image and the correct category appeared in the ﬁrst ﬁve entries of this list for all but 15.3% of the test examples. Since then, these competitions are consistently won by deep convolutional nets, and as of this writing, advances in deep learning have brought the latest top-5 error rate in this contest down to 3.6%, as shown in Fig. 1.12. Deep learning has also had a dramatic impact on speech recognition. After improving throughout the 1990s, the error rates for speech recognition stagnated starting in about 2000. The introduction of deep learning (Dahl et al., 2010; Deng et al., 2010b; Seide et al., 2011; Hinton et al., 2012a) to speech recognition resulted in a sudden drop of error rates, with some error rates cut in half. We will explore this history in more detail in Sec. 12.3. Deep networks have also had spectacular successes for pedestrian detection and image segmentation (Sermanet et al., 2013; Farabet et al., 2013; Couprie et al., 2013) and yielded superhuman performance in traﬃc sign classiﬁcation (Ciresan 23

CHAPTER 1. INTRODUCTION

Figure 1.10: Initially, the number of connections between neurons in artiﬁcial neural networks was limited by hardware capabilities. Today, the number of connections between neurons is mostly a design consideration. Some artiﬁcial neural networks have nearly as many connections per neuron as a cat, and it is quite common for other neural networks to have as many connections per neuron as smaller mammals like mice. Even the human brain does not have an exorbitant amount of connections per neuron. Biological neural network sizes from Wikipedia (2015). 1. Adaptive linear element (Widrow and Hoﬀ, 1960) 2. Neocognitron (Fukushima, 1980) 3. GPU-accelerated convolutional network (Chellapilla et al., 2006) 4. Deep Boltzmann machine (Salakhutdinov and Hinton, 2009a) 5. Unsupervised convolutional network (Jarrett et al., 2009) 6. GPU-accelerated multilayer perceptron (Ciresan et al., 2010) 7. Distributed autoencoder (Le et al., 2012) 8. Multi-GPU convolutional network (Krizhevsky et al., 2012) 9. COTS HPC unsupervised convolutional network (Coates et al., 2013) 10. GoogLeNet (Szegedy et al., 2014a)

24

CHAPTER 1. INTRODUCTION

et al., 2012). At the same time that the scale and accuracy of deep networks has increased, so has the complexity of the tasks that they can solve. Goodfellow et al. (2014d) showed that neural networks could learn to output an entire sequence of characters transcribed from an image, rather than just identifying a single object. Previously, it was widely believed that this kind of learning required labeling of the individual elements of the sequence (Gülçehre and Bengio, 2013). Recurrent neural networks, such as the LSTM sequence model mentioned above, are now used to model relationships between sequences and other sequences rather than just ﬁxed inputs. This sequence-to-sequence learning seems to be on the cusp of revolutionizing another application: machine translation (Sutskever et al., 2014; Bahdanau et al., 2015). This trend of increasing complexity has been pushed to its logical conclusion with the introduction of neural Turing machines (Graves et al., 2014a) that learn to read from memory cells and write arbitrary content to memory cells. Such neural networks can learn simple programs from examples of desired behavior. For example, they can learn to sort lists of numbers given examples of scrambled and sorted sequences. This self-programming technology is in its infancy, but in the future could in principle be applied to nearly any task. Another crowning achievement of deep learning is its extension to the domain of reinforcement learning. In the context of reinforcement learning, an autonomous agent must learn to perform a task by trial and error, without any guidance from the human operator. DeepMind demonstrated that a reinforcement learning system based on deep learning is capable of learning to play Atari video games, reaching human-level performance on many tasks (Mnih et al., 2015). Deep learning has also signiﬁcantly improved the performance of reinforcement learning for robotics (Finn et al., 2015). Many of these applications of deep learning are highly proﬁtable. Deep learning is now used by many top technology companies including Google, Microsoft, Facebook, IBM, Baidu, Apple, Adobe, Netﬂix, NVIDIA and NEC. Advances in deep learning have also depended heavily on advances in software infrastructure. Software libraries such as Theano (Bergstra et al., 2010; Bastien et al., 2012), PyLearn2 (Goodfellow et al., 2013c), Torch (Collobert et al., 2011b), DistBelief (Dean et al., 2012), Caﬀe (Jia, 2013), MXNet (Chen et al., 2015), and TensorFlow (Abadi et al., 2015) have all supported important research projects or commercial products. Deep learning has also made contributions back to other sciences. Modern convolutional networks for object recognition provide a model of visual processing 25

CHAPTER 1. INTRODUCTION

that neuroscientists can study (DiCarlo, 2013). Deep learning also provides useful tools for processing massive amounts of data and making useful predictions in scientiﬁc ﬁelds. It has been successfully used to predict how molecules will interact in order to help pharmaceutical companies design new drugs (Dahl et al., 2014), to search for subatomic particles (Baldi et al., 2014), and to automatically parse microscope images used to construct a 3-D map of the human brain (KnowlesBarley et al., 2014). We expect deep learning to appear in more and more scientiﬁc ﬁelds in the future. In summary, deep learning is an approach to machine learning that has drawn heavily on our knowledge of the human brain, statistics and applied math as it developed over the past several decades. In recent years, it has seen tremendous growth in its popularity and usefulness, due in large part to more powerful computers, larger datasets and techniques to train deeper networks. The years ahead are full of challenges and opportunities to improve deep learning even further and bring it to new frontiers.

26

Number of neurons (logarithmic scale)

CHAPTER 1. INTRODUCTION

1011 1010 109 108 107 106 105 104 103 102 101 100 10−1 10−2

Increasing neural network size over time Human 17

8

19

16

20

Octopus

18

14 11

Frog Bee

3

Ant Leech

13 1

2 4

1950

12

6

1985

2000

5

9 7

15

Roundworm

10

2015

2056

Sponge

Figure 1.11: Since the introduction of hidden units, artiﬁcial neural networks have doubled in size roughly every 2.4 years. Biological neural network sizes from Wikipedia (2015). 1. Perceptron (Rosenblatt, 1958, 1962) 2. Adaptive linear element (Widrow and Hoﬀ, 1960) 3. Neocognitron (Fukushima, 1980) 4. Early back-propagation network (Rumelhart et al., 1986b) 5. Recurrent neural network for speech recognition (Robinson and Fallside, 1991) 6. Multilayer perceptron for speech recognition (Bengio et al., 1991) 7. Mean ﬁeld sigmoid belief network (Saul et al., 1996) 8. LeNet-5 (LeCun et al., 1998b) 9. Echo state network (Jaeger and Haas, 2004) 10. Deep belief network (Hinton et al., 2006) 11. GPU-accelerated convolutional network (Chellapilla et al., 2006) 12. Deep Boltzmann machine (Salakhutdinov and Hinton, 2009a) 13. GPU-accelerated deep belief network (Raina et al., 2009) 14. Unsupervised convolutional network (Jarrett et al., 2009) 15. GPU-accelerated multilayer perceptron (Ciresan et al., 2010) 16. OMP-1 network (Coates and Ng, 2011) 17. Distributed autoencoder (Le et al., 2012) 18. Multi-GPU convolutional network (Krizhevsky et al., 2012) 19. COTS HPC unsupervised convolutional network (Coates et al., 2013) 20. GoogLeNet (Szegedy et al., 2014a)

27

ILSVRC classiﬁcation error rate

CHAPTER 1. INTRODUCTION

0.30

Decreasing error rate over time

0.25 0.20 0.15 0.10 0.05 0.00 2010

2011

2012

2013

2014

2015

Year

Figure 1.12: Since deep networks reached the scale necessary to compete in the ImageNet Large Scale Visual Recognition Challenge, they have consistently won the competition every year, and yielded lower and lower error rates each time. Data from Russakovsky et al. (2014b) and He et al. (2015).

28

Part I

Applied Math and Machine Learning Basics

29

This part of the book introduces the basic mathematical concepts needed to understand deep learning. We begin with general ideas from applied math that allow us to deﬁne functions of many variables, ﬁnd the highest and lowest points on these functions and quantify degrees of belief. Next, we describe the fundamental goals of machine learning. We describe how to accomplish these goals by specifying a model that represents certain beliefs, designing a cost function that measures how well those beliefs correspond with reality and using a training algorithm to minimize that cost function. This elementary framework is the basis for a broad variety of machine learning algorithms, including approaches to machine learning that are not deep. In the subsequent parts of the book, we develop deep learning algorithms within this framework.

30

Chapter 2

Linear Algebra Linear algebra is a branch of mathematics that is widely used throughout science and engineering. However, because linear algebra is a form of continuous rather than discrete mathematics, many computer scientists have little experience with it. A good understanding of linear algebra is essential for understanding and working with many machine learning algorithms, especially deep learning algorithms. We therefore precede our introduction to deep learning with a focused presentation of the key linear algebra prerequisites. If you are already familiar with linear algebra, feel free to skip this chapter. If you have previous experience with these concepts but need a detailed reference sheet to review key formulas, we recommend The Matrix Cookbook (Petersen and Pedersen, 2006). If you have no exposure at all to linear algebra, this chapter will teach you enough to read this book, but we highly recommend that you also consult another resource focused exclusively on teaching linear algebra, such as Shilov (1977). This chapter will completely omit many important linear algebra topics that are not essential for understanding deep learning.

2.1

Scalars, Vectors, Matrices and Tensors

The study of linear algebra involves several types of mathematical objects: • Scalars: A scalar is just a single number, in contrast to most of the other objects studied in linear algebra, which are usually arrays of multiple numbers. We write scalars in italics. We usually give scalars lower-case variable names. When we introduce them, we specify what kind of number they are. For 31

CHAPTER 2. LINEAR ALGEBRA

example, we might say “Let s ∈ R be the slope of the line,” while deﬁning a real-valued scalar, or “Let n ∈ N be the number of units,” while deﬁning a natural number scalar. • Vectors: A vector is an array of numbers. The numbers are arranged in order. We can identify each individual number by its index in that ordering. Typically we give vectors lower case names written in bold typeface, such as x. The elements of the vector are identiﬁed by writing its name in italic typeface, with a subscript. The ﬁrst element of x is x1, the second element is x 2 and so on. We also need to say what kind of numbers are stored in the vector. If each element is in R, and the vector has n elements, then the vector lies in the set formed by taking the Cartesian product of R n times, denoted as R n. When we need to explicitly identify the elements of a vector, we write them as a column enclosed in square brackets: x1 x2 x = .. . (2.1) . xn

We can think of vectors as identifying points in space, with each element giving the coordinate along a diﬀerent axis. Sometimes we need to index a set of elements of a vector. In this case, we deﬁne a set containing the indices and write the set as a subscript. For example, to access x 1, x3 and x6 , we deﬁne the set S = {1, 3, 6} and write xS . We use the − sign to index the complement of a set. For example x−1 is the vector containing all elements of x except for x1 , and x−S is the vector containing all of the elements of x except for x 1, x3 and x6 . • Matrices: A matrix is a 2-D array of numbers, so each element is identiﬁed by two indices instead of just one. We usually give matrices upper-case variable names with bold typeface, such as A. If a real-valued matrix A has a height of m and a width of n, then we say that A ∈ R m×n. We usually identify the elements of a matrix using its name in italic but not bold font, and the indices are listed with separating commas. For example, A1,1 is the upper left entry of A and Am,n is the bottom right entry. We can identify all of the numbers with vertical coordinate i by writing a “ :” for the horizontal coordinate. For example, Ai,: denotes the horizontal cross section of A with vertical coordinate i. This is known as the i-th row of A. Likewise, A:,i is 32

CHAPTER 2. LINEAR ALGEBRA

A1,1 A = A2,1 A3,1

A1,2 A1,1 A2,2 ⇒ A = A1,2 A3,2

A2,1 A2,2

A3,1 A3,2

Figure 2.1: The transpose of the matrix can be thought of as a mirror image across the main diagonal.

the i-th column of A. When we need to explicitly identify the elements of a matrix, we write them as an array enclosed in square brackets: A1,1 A 1,2 . (2.2) A2,1 A 2,2 Sometimes we may need to index matrix-valued expressions that are not just a single letter. In this case, we use subscripts after the expression, but do not convert anything to lower case. For example, f (A)i,j gives element (i, j) of the matrix computed by applying the function f to A. • Tensors: In some cases we will need an array with more than two axes. In the general case, an array of numbers arranged on a regular grid with a variable number of axes is known as a tensor. We denote a tensor named “A” with this typeface: A. We identify the element of A at coordinates (i, j, k) by writing A i,j,k. One important operation on matrices is the transpose. The transpose of a matrix is the mirror image of the matrix across a diagonal line, called the main diagonal, running down and to the right, starting from its upper left corner. See Fig. 2.1 for a graphical depiction of this operation. We denote the transpose of a matrix A as A , and it is deﬁned such that (A)i,j = Aj,i .

(2.3)

Vectors can be thought of as matrices that contain only one column. The transpose of a vector is therefore a matrix with only one row. Sometimes we 33

CHAPTER 2. LINEAR ALGEBRA

deﬁne a vector by writing out its elements in the text inline as a row matrix, then using the transpose operator to turn it into a standard column vector, e.g., x = [x1, x2, x3 ] . A scalar can be thought of as a matrix with only a single entry. From this, we can see that a scalar is its own transpose: a = a . We can add matrices to each other, as long as they have the same shape, just by adding their corresponding elements: C = A + B where Ci,j = Ai,j + B i,j . We can also add a scalar to a matrix or multiply a matrix by a scalar, just by performing that operation on each element of a matrix: D = a · B + c where Di,j = a · Bi,j + c.

In the context of deep learning, we also use some less conventional notation. We allow the addition of matrix and a vector, yielding another matrix: C = A + b, where Ci,j = Ai,j + b j. In other words, the vector b is added to each row of the matrix. This shorthand eliminates the need to deﬁne a matrix with b copied into each row before doing the addition. This implicit copying of b to many locations is called broadcasting.

2.2

Multiplying Matrices and Vectors

One of the most important operations involving matrices is multiplication of two matrices. The matrix product of matrices A and B is a third matrix C . In order for this product to be deﬁned, A must have the same number of columns as B has rows. If A is of shape m × n and B is of shape n × p, then C is of shape m × p. We can write the matrix product just by placing two or more matrices together, e.g. C = AB . (2.4) The product operation is deﬁned by Ci,j = Ai,k B k,j.

(2.5)

k

Note that the standard product of two matrices is not just a matrix containing the product of the individual elements. Such an operation exists and is called the element-wise product or Hadamard product, and is denoted as A B .

The dot product between two vectors x and y of the same dimensionality is the matrix product xy. We can think of the matrix product C = AB as computing Ci,j as the dot product between row i of A and column j of B . 34

CHAPTER 2. LINEAR ALGEBRA

Matrix product operations have many useful properties that make mathematical analysis of matrices more convenient. For example, matrix multiplication is distributive: A(B + C ) = AB + AC . (2.6) It is also associative: A(BC ) = (AB )C .

(2.7)

Matrix multiplication is not commutative (the condition AB = BA does not always hold), unlike scalar multiplication. However, the dot product between two vectors is commutative: xy = y x. (2.8) The transpose of a matrix product has a simple form: (AB ) = B A .

(2.9)

This allows us to demonstrate Eq. 2.8, by exploiting the fact that the value of such a product is a scalar and therefore equal to its own transpose: x y= x y = y x. (2.10)

Since the focus of this textbook is not linear algebra, we do not attempt to develop a comprehensive list of useful properties of the matrix product here, but the reader should be aware that many more exist. We now know enough linear algebra notation to write down a system of linear equations: Ax = b (2.11)

where A ∈ R m×n is a known matrix, b ∈ Rm is a known vector, and x ∈ Rn is a vector of unknown variables we would like to solve for. Each element xi of x is one of these unknown variables. Each row of A and each element of b provide another constraint. We can rewrite Eq. 2.11 as: A1,: x = b1

(2.12)

A2,: x = b2

(2.13)

...

(2.14)

Am,: x = bm

(2.15)

A1,1 x1 + A1,2 x2 + · · · + A 1,nxn = b1

(2.16)

or, even more explicitly, as:

35

CHAPTER 2. LINEAR ALGEBRA

1 0 0 0 1 0 0 0 1 Figure 2.2: Example identity matrix: This is I3 .

A2,1 x1 + A2,2 x2 + · · · + A 2,nxn = b2

(2.17)

...

(2.18)

A m,1x 1 + Am,2x 2 + · · · + A m,nxn = bm .

(2.19)

Matrix-vector product notation provides a more compact representation for equations of this form.

2.3

Identity and Inverse Matrices

Linear algebra oﬀers a powerful tool called matrix inversion that allows us to analytically solve Eq. 2.11 for many values of A. To describe matrix inversion, we ﬁrst need to deﬁne the concept of an identity matrix. An identity matrix is a matrix that does not change any vector when we multiply that vector by that matrix. We denote the identity matrix that preserves n-dimensional vectors as I n. Formally, I n ∈ Rn×n , and ∀x ∈ Rn, I nx = x.

(2.20)

The structure of the identity matrix is simple: all of the entries along the main diagonal are 1, while all of the other entries are zero. See Fig. 2.2 for an example. The matrix inverse of A is denoted as A−1, and it is deﬁned as the matrix such that A−1 A = In . (2.21) We can now solve Eq. 2.11 by the following steps: Ax = b

(2.22)

A−1 Ax = A −1b

(2.23)

In x = A−1 b

(2.24)

36

CHAPTER 2. LINEAR ALGEBRA

x = A−1 b.

(2.25)

Of course, this depends on it being possible to ﬁnd A−1 . We discuss the conditions for the existence of A−1 in the following section. When A−1 exists, several diﬀerent algorithms exist for ﬁnding it in closed form. In theory, the same inverse matrix can then be used to solve the equation many times for diﬀerent values of b. However, A −1 is primarily useful as a theoretical tool, and should not actually be used in practice for most software applications. Because A−1 can be represented with only limited precision on a digital computer, algorithms that make use of the value of b can usually obtain more accurate estimates of x.

2.4

Linear Dependence and Span

In order for A−1 to exist, Eq. 2.11 must have exactly one solution for every value of b. However, it is also possible for the system of equations to have no solutions or inﬁnitely many solutions for some values of b. It is not possible to have more than one but less than inﬁnitely many solutions for a particular b; if both x and y are solutions then z = αx + (1 − α)y (2.26) is also a solution for any real α. To analyze how many solutions the equation has, we can think of the columns of A as specifying diﬀerent directions we can travel from the origin (the point speciﬁed by the vector of all zeros), and determine how many ways there are of reaching b. In this view, each element of x speciﬁes how far we should travel in each of these directions, with xi specifying how far to move in the direction of column i: Ax = x iA :,i. (2.27) i

In general, this kind of operation is called a linear combination. Formally, a linear combination of some set of vectors {v(1) , . . . , v(n) } is given by multiplying each vector v (i) by a corresponding scalar coeﬃcient and adding the results: ci v (i). (2.28) i

The span of a set of vectors is the set of all points obtainable by linear combination of the original vectors. 37

CHAPTER 2. LINEAR ALGEBRA

Determining whether Ax = b has a solution thus amounts to testing whether b is in the span of the columns of A. This particular span is known as the column space or the range of A. In order for the system Ax = b to have a solution for all values of b ∈ R m, we therefore require that the column space of A be all of Rm . If any point in R m is excluded from the column space, that point is a potential value of b that has no solution. The requirement that the column space of A be all of R m implies immediately that A must have at least m columns, i.e., n ≥ m. Otherwise, the dimensionality of the column space would be less than m. For example, consider a 3 × 2 matrix. The target b is 3-D, but x is only 2-D, so modifying the value of x at best allows us to trace out a 2-D plane within R 3. The equation has a solution if and only if b lies on that plane. Having n ≥ m is only a necessary condition for every point to have a solution. It is not a suﬃcient condition, because it is possible for some of the columns to be redundant. Consider a 2 ×2 matrix where both of the columns are identical. This has the same column space as a 2 × 1 matrix containing only one copy of the replicated column. In other words, the column space is still just a line, and fails to encompass all of R2 , even though there are two columns. Formally, this kind of redundancy is known as linear dependence. A set of vectors is linearly independent if no vector in the set is a linear combination of the other vectors. If we add a vector to a set that is a linear combination of the other vectors in the set, the new vector does not add any points to the set’s span. This means that for the column space of the matrix to encompass all of Rm, the matrix must contain at least one set of m linearly independent columns. This condition is both necessary and suﬃcient for Eq. 2.11 to have a solution for every value of b. Note that the requirement is for a set to have exactly m linear independent columns, not at least m. No set of m-dimensional vectors can have more than m mutually linearly independent columns, but a matrix with more than m columns may have more than one such set. In order for the matrix to have an inverse, we additionally need to ensure that Eq. 2.11 has at most one solution for each value of b. To do so, we need to ensure that the matrix has at most m columns. Otherwise there is more than one way of parametrizing each solution. Together, this means that the matrix must be square, that is, we require that m = n and that all of the columns must be linearly independent. A square matrix with linearly dependent columns is known as singular. If A is not square or is square but singular, it can still be possible to solve the 38

CHAPTER 2. LINEAR ALGEBRA

equation. However, we can not use the method of matrix inversion to ﬁnd the solution. So far we have discussed matrix inverses as being multiplied on the left. It is also possible to deﬁne an inverse that is multiplied on the right: AA−1 = I.

(2.29)

For square matrices, the left inverse and right inverse are equal.

2.5

Norms

Sometimes we need to measure the size of a vector. In machine learning, we usually measure the size of vectors using a function called a norm. Formally, the Lp norm is given by 1 p (2.30) ||x||p = |x i|p i

for p ∈ R, p ≥ 1. Norms, including the L p norm, are functions mapping vectors to non-negative values. On an intuitive level, the norm of a vector x measures the distance from the origin to the point x. More rigorously, a norm is any function f that satisﬁes the following properties: • f (x) = 0 ⇒ x = 0 • f (x + y) ≤ f (x) + f (y ) (the triangle inequality) • ∀α ∈ R, f (αx) = |α|f(x) The L2 norm, with p = 2, is known as the Euclidean norm. It is simply the Euclidean distance from the origin to the point identiﬁed by x. The L 2 norm is used so frequently in machine learning that it is often denoted simply as ||x||, with the subscript 2 omitted. It is also common to measure the size of a vector using the squared L2 norm, which can be calculated simply as x x.

The squared L 2 norm is more convenient to work with mathematically and computationally than the L 2 norm itself. For example, the derivatives of the squared L2 norm with respect to each element of x each depend only on the corresponding element of x, while all of the derivatives of the L2 norm depend on the entire vector. In many contexts, the squared L 2 norm may be undesirable 39

CHAPTER 2. LINEAR ALGEBRA

because it increases very slowly near the origin. In several machine learning applications, it is important to discriminate between elements that are exactly zero and elements that are small but nonzero. In these cases, we turn to a function that grows at the same rate in all locations, but retains mathematical simplicity: the L1 norm. The L1 norm may be simpliﬁed to ||x||1 = |xi |. (2.31) i

The L 1 norm is commonly used in machine learning when the diﬀerence between zero and nonzero elements is very important. Every time an element of x moves away from 0 by , the L1 norm increases by . We sometimes measure the size of the vector by counting its number of nonzero elements. Some authors refer to this function as the “L 0 norm,” but this is incorrect terminology. The number of non-zero entries in a vector is not a norm, because scaling the vector by α does not change the number of nonzero entries. The L1 norm is often used as a substitute for the number of nonzero entries. One other norm that commonly arises in machine learning is the L∞ norm, also known as the max norm. This norm simpliﬁes to the absolute value of the element with the largest magnitude in the vector, ||x||∞ = max |xi |. i

(2.32)

Sometimes we may also wish to measure the size of a matrix. In the context of deep learning, the most common way to do this is with the otherwise obscure Frobenius norm A 2i,j , (2.33) ||A|| F = i,j

which is analogous to the L 2 norm of a vector. The dot product of two vectors can be rewritten in terms of norms. Speciﬁcally, x y = ||x||2||y|| 2 cos θ where θ is the angle between x and y .

2.6

Special Kinds of Matrices and Vectors

Some special kinds of matrices and vectors are particularly useful. 40

(2.34)

CHAPTER 2. LINEAR ALGEBRA

Diagonal matrices consist mostly of zeros and have non-zero entries only along the main diagonal. Formally, a matrix D is diagonal if and only if Di,j = 0 for = j . We have already seen one example of a diagonal matrix: the identity all i matrix, where all of the diagonal entries are 1. We write diag(v) to denote a square diagonal matrix whose diagonal entries are given by the entries of the vector v. Diagonal matrices are of interest in part because multiplying by a diagonal matrix is very computationally eﬃcient. To compute diag(v)x, we only need to scale each element xi by vi . In other words, diag( v)x = v x. Inverting a square diagonal matrix is also eﬃcient. The inverse exists only if every diagonal entry is nonzero, and in that case, diag(v)−1 = diag([1/v 1, . . . , 1/vn ]). In many cases, we may derive some very general machine learning algorithm in terms of arbitrary matrices, but obtain a less expensive (and less descriptive) algorithm by restricting some matrices to be diagonal. Not all diagonal matrices need be square. It is possible to construct a rectangular diagonal matrix. Non-square diagonal matrices do not have inverses but it is still possible to multiply by them cheaply. For a non-square diagonal matrix D, the product Dx will involve scaling each element of x, and either concatenating some zeros to the result if D is taller than it is wide, or discarding some of the last elements of the vector if D is wider than it is tall. A symmetric matrix is any matrix that is equal to its own transpose: A = A .

(2.35)

Symmetric matrices often arise when the entries are generated by some function of two arguments that does not depend on the order of the arguments. For example, if A is a matrix of distance measurements, with Ai,j giving the distance from point i to point j , then Ai,j = Aj,i because distance functions are symmetric. A unit vector is a vector with unit norm: ||x||2 = 1.

(2.36)

A vector x and a vector y are orthogonal to each other if xy = 0. If both vectors have nonzero norm, this means that they are at a 90 degree angle to each other. In Rn , at most n vectors may be mutually orthogonal with nonzero norm. If the vectors are not only orthogonal but also have unit norm, we call them orthonormal. An orthogonal matrix is a square matrix whose rows are mutually orthonormal and whose columns are mutually orthonormal: A A = AA = I. 41

(2.37)

CHAPTER 2. LINEAR ALGEBRA

This implies that A−1 = A ,

(2.38)

so orthogonal matrices are of interest because their inverse is very cheap to compute. Pay careful attention to the deﬁnition of orthogonal matrices. Counterintuitively, their rows are not merely orthogonal but fully orthonormal. There is no special term for a matrix whose rows or columns are orthogonal but not orthonormal.

2.7

Eigendecomposition

Many mathematical objects can be understood better by breaking them into constituent parts, or ﬁnding some properties of them that are universal, not caused by the way we choose to represent them. For example, integers can be decomposed into prime factors. The way we represent the number 12 will change depending on whether we write it in base ten or in binary, but it will always be true that 12 = 2 × 2× 3. From this representation we can conclude useful properties, such as that 12 is not divisible by 5, or that any integer multiple of 12 will be divisible by 3. Much as we can discover something about the true nature of an integer by decomposing it into prime factors, we can also decompose matrices in ways that show us information about their functional properties that is not obvious from the representation of the matrix as an array of elements. One of the most widely used kinds of matrix decomposition is called eigendecomposition, in which we decompose a matrix into a set of eigenvectors and eigenvalues. An eigenvector of a square matrix A is a non-zero vector v such that multiplication by A alters only the scale of v: Av = λv.

(2.39)

The scalar λ is known as the eigenvalue corresponding to this eigenvector. (One can also ﬁnd a left eigenvector such that v A = λv , but we are usually concerned with right eigenvectors). If v is an eigenvector of A, then so is any rescaled vector sv for s ∈ R, s = 0. Moreover, sv still has the same eigenvalue. For this reason, we usually only look for unit eigenvectors. Suppose that a matrix A has n linearly independent eigenvectors, {v (1) , . . . , v(n)}, with corresponding eigenvalues {λ1, . . . , λ n }. We may concatenate all of the 42

CHAPTER 2. LINEAR ALGEBRA

Figure 2.3: An example of the eﬀect of eigenvectors and eigenvalues. Here, we have a matrix A with two orthonormal eigenvectors, v (1) with eigenvalue λ1 and v (2) with eigenvalue λ2 . (Left) We plot the set of all unit vectors u ∈ R 2 as a unit circle. (Right) We plot the set of all points Au. By observing the way that A distorts the unit circle, we can see that it scales space in direction v(i) by λi.

eigenvectors to form a matrix V with one eigenvector per column: V = [v(1) , . . . , v(n)]. Likewise, we can concatenate the eigenvalues to form a vector λ = [λ 1 , . . . , λn ]. The eigendecomposition of A is then given by A = V diag(λ)V −1 .

(2.40)

We have seen that constructing matrices with speciﬁc eigenvalues and eigenvectors allows us to stretch space in desired directions. However, we often want to decompose matrices into their eigenvalues and eigenvectors. Doing so can help us to analyze certain properties of the matrix, much as decomposing an integer into its prime factors can help us understand the behavior of that integer. Not every matrix can be decomposed into eigenvalues and eigenvectors. In some 43

CHAPTER 2. LINEAR ALGEBRA

cases, the decomposition exists, but may involve complex rather than real numbers. Fortunately, in this book, we usually need to decompose only a speciﬁc class of matrices that have a simple decomposition. Speciﬁcally, every real symmetric matrix can be decomposed into an expression using only real-valued eigenvectors and eigenvalues: A = QΛQ, (2.41) where Q is an orthogonal matrix composed of eigenvectors of A, and Λ is a diagonal matrix. The eigenvalue Λi,i is associated with the eigenvector in column i of Q, denoted as Q:,i. Because Q is an orthogonal matrix, we can think of A as scaling space by λi in direction v (i). See Fig. 2.3 for an example. While any real symmetric matrix A is guaranteed to have an eigendecomposition, the eigendecomposition may not be unique. If any two or more eigenvectors share the same eigenvalue, then any set of orthogonal vectors lying in their span are also eigenvectors with that eigenvalue, and we could equivalently choose a Q using those eigenvectors instead. By convention, we usually sort the entries of Λ in descending order. Under this convention, the eigendecomposition is unique only if all of the eigenvalues are unique. The eigendecomposition of a matrix tells us many useful facts about the matrix. The matrix is singular if and only if any of the eigenvalues are zero. The eigendecomposition of a real symmetric matrix can also be used to optimize quadratic expressions of the form f( x) = x Ax subject to ||x||2 = 1. Whenever x is equal to an eigenvector of A, f takes on the value of the corresponding eigenvalue. The maximum value of f within the constraint region is the maximum eigenvalue and its minimum value within the constraint region is the minimum eigenvalue. A matrix whose eigenvalues are all positive is called positive deﬁnite. A matrix whose eigenvalues are all positive or zero-valued is called positive semideﬁnite. Likewise, if all eigenvalues are negative, the matrix is negative deﬁnite, and if all eigenvalues are negative or zero-valued, it is negative semideﬁnite. Positive semideﬁnite matrices are interesting because they guarantee that ∀x, x Ax ≥ 0. Positive deﬁnite matrices additionally guarantee that x Ax = 0 ⇒ x = 0.

2.8

Singular Value Decomposition

In Sec. 2.7, we saw how to decompose a matrix into eigenvectors and eigenvalues. The singular value decomposition (SVD) provides another way to factorize a matrix, into singular vectors and singular values. The SVD allows us to discover some of the same kind of information as the eigendecomposition. However, the SVD is 44

CHAPTER 2. LINEAR ALGEBRA

more generally applicable. Every real matrix has a singular value decomposition, but the same is not true of the eigenvalue decomposition. For example, if a matrix is not square, the eigendecomposition is not deﬁned, and we must use a singular value decomposition instead. Recall that the eigendecomposition involves analyzing a matrix A to discover a matrix V of eigenvectors and a vector of eigenvalues λ such that we can rewrite A as A = V diag(λ)V −1 . (2.42) The singular value decomposition is similar, except this time we will write A as a product of three matrices: A = U DV .

(2.43)

Suppose that A is an m × n matrix. Then U is deﬁned to be an m × m matrix, D to be an m × n matrix, and V to be an n × n matrix. Each of these matrices is deﬁned to have a special structure. The matrices U and V are both deﬁned to be orthogonal matrices. The matrix D is deﬁned to be a diagonal matrix. Note that D is not necessarily square. The elements along the diagonal of D are known as the singular values of the matrix A. The columns of U are known as the left-singular vectors. The columns of V are known as as the right-singular vectors. We can actually interpret the singular value decomposition of A in terms of the eigendecomposition of functions of A. The left-singular vectors of A are the eigenvectors of AA . The right-singular vectors of A are the eigenvectors of A A. The non-zero singular values of A are the square roots of the eigenvalues of A A. The same is true for AA . Perhaps the most useful feature of the SVD is that we can use it to partially generalize matrix inversion to non-square matrices, as we will see in the next section.

2.9

The Moore-Penrose Pseudoinverse

Matrix inversion is not deﬁned for matrices that are not square. Suppose we want to make a left-inverse B of a matrix A, so that we can solve a linear equation Ax = y 45

(2.44)

CHAPTER 2. LINEAR ALGEBRA

by left-multiplying each side to obtain x = By .

(2.45)

Depending on the structure of the problem, it may not be possible to design a unique mapping from A to B . If A is taller than it is wide, then it is possible for this equation to have no solution. If A is wider than it is tall, then there could be multiple possible solutions. The Moore-Penrose pseudoinverse allows us to make some headway in these cases. The pseudoinverse of A is deﬁned as a matrix A+ = lim(A A + αI ) −1 A . α0

(2.46)

Practical algorithms for computing the pseudoinverse are not based on this deﬁnition, but rather the formula A + = V D + U ,

(2.47)

where U, D and V are the singular value decomposition of A, and the pseudoinverse D+ of a diagonal matrix D is obtained by taking the reciprocal of its non-zero elements then taking the transpose of the resulting matrix. When A has more columns than rows, then solving a linear equation using the pseudoinverse provides one of the many possible solutions. Speciﬁcally, it provides the solution x = A+ y with minimal Euclidean norm ||x||2 among all possible solutions. When A has more rows than columns, it is possible for there to be no solution. In this case, using the pseudoinverse gives us the x for which Ax is as close as possible to y in terms of Euclidean norm ||Ax − y||2 .

2.10

The Trace Operator

The trace operator gives the sum of all of the diagonal entries of a matrix: Tr(A) = Ai,i . (2.48) i

The trace operator is useful for a variety of reasons. Some operations that are diﬃcult to specify without resorting to summation notation can be speciﬁed using 46

CHAPTER 2. LINEAR ALGEBRA

matrix products and the trace operator. For example, the trace operator provides an alternative way of writing the Frobenius norm of a matrix: (2.49) ||A|| F = Tr(AA). Writing an expression in terms of the trace operator opens up opportunities to manipulate the expression using many useful identities. For example, the trace operator is invariant to the transpose operator: Tr(A) = Tr(A ).

(2.50)

The trace of a square matrix composed of many factors is also invariant to moving the last factor into the ﬁrst position, if the shapes of the corresponding matrices allow the resulting product to be deﬁned: Tr(ABC ) = Tr(CAB) = Tr(BCA)

(2.51)

or more generally, Tr(

n

F

(i)

) = Tr(F

i=1

(n)

n−1

F (i) ).

(2.52)

i=1

This invariance to cyclic permutation holds even if the resulting product has a diﬀerent shape. For example, for A ∈ Rm×n and B ∈ R n×m, we have Tr(AB ) = Tr(BA)

(2.53)

even though AB ∈ Rm×m and BA ∈ Rn×n .

Another useful fact to keep in mind is that a scalar is its own trace: a = Tr(a).

2.11

The Determinant

The determinant of a square matrix, denoted det(A), is a function mapping matrices to real scalars. The determinant is equal to the product of all the eigenvalues of the matrix. The absolute value of the determinant can be thought of as a measure of how much multiplication by the matrix expands or contracts space. If the determinant is 0, then space is contracted completely along at least one dimension, causing it to lose all of its volume. If the determinant is 1, then the transformation is volume-preserving.

47

CHAPTER 2. LINEAR ALGEBRA

2.12

Example: Principal Components Analysis

One simple machine learning algorithm, principal components analysis or PCA can be derived using only knowledge of basic linear algebra. Suppose we have a collection of m points {x(1) , . . . , x(m)} in Rn . Suppose we would like to apply lossy compression to these points. Lossy compression means storing the points in a way that requires less memory but may lose some precision. We would like to lose as little precision as possible. One way we can encode these points is to represent a lower-dimensional version of them. For each point x(i) ∈ R n we will ﬁnd a corresponding code vector c (i) ∈ R l. If l is smaller than n, it will take less memory to store the code points than the original data. We will want to ﬁnd some encoding function that produces the code for an input, f (x) = c, and a decoding function that produces the reconstructed input given its code, x ≈ g(f (x)). PCA is deﬁned by our choice of the decoding function. Speciﬁcally, to make the decoder very simple, we choose to use matrix multiplication to map the code back into Rn . Let g(c) = Dc, where D ∈ Rn×l is the matrix deﬁning the decoding. Computing the optimal code for this decoder could be a diﬃcult problem. To keep the encoding problem easy, PCA constrains the columns of D to be orthogonal to each other. (Note that D is still not technically “an orthogonal matrix” unless l = n)

With the problem as described so far, many solutions are possible, because we can increase the scale of D:,i if we decrease c i proportionally for all points. To give the problem a unique solution, we constrain all of the columns of D to have unit norm. In order to turn this basic idea into an algorithm we can implement, the ﬁrst thing we need to do is ﬁgure out how to generate the optimal code point c∗ for each input point x. One way to do this is to minimize the distance between the input point x and its reconstruction, g( c∗). We can measure this distance using a norm. In the principal components algorithm, we use the L2 norm: c ∗ = arg min ||x − g(c)|| 2.

(2.54)

c

We can switch to the squared L 2 norm instead of the L2 norm itself, because both are minimized by the same value of c . This is because the L 2 norm is nonnegative and the squaring operation is monotonically increasing for non-negative

48

CHAPTER 2. LINEAR ALGEBRA

arguments. c ∗ = arg min ||x − g(c)|| 22.

(2.55)

c

The function being minimized simpliﬁes to (x − g(c)) (x − g(c))

(2.56)

(by the deﬁnition of the L2 norm, Eq. 2.30) = x x − xg(c) − g (c) x + g(c)g(c)

(2.57)

(by the distributive property) = x x − 2x g(c) + g (c) g(c)

(2.58)

(because the scalar g(x) x is equal to the transpose of itself). We can now change the function being minimized again, to omit the ﬁrst term, since this term does not depend on c: c∗ = arg min −2x g(c) + g (c) g(c).

(2.59)

c

To make further progress, we must substitute in the deﬁnition of g(c): c∗ = arg min −2x Dc + c D Dc

(2.60)

= arg min −2x Dc + c Il c

(2.61)

c

c

(by the orthogonality and unit norm constraints on D) = arg min −2x Dc + c c

(2.62)

c

We can solve this optimization problem using vector calculus (see Sec. 4.3 if you do not know how to do this): ∇c(−2xDc + c c) = 0

(2.63)

− 2Dx + 2c = 0

(2.64)

c = D x.

(2.65)

This makes the algorithm eﬃcient: we can optimally encode x just using a matrix-vector operation. To encode a vector, we apply the encoder function f(x) = D x. 49

(2.66)

CHAPTER 2. LINEAR ALGEBRA

Using a further matrix multiplication, we can also deﬁne the PCA reconstruction operation: r(x) = g (f (x)) = DD x. (2.67) Next, we need to choose the encoding matrix D. To do so, we revisit the idea of minimizing the L2 distance between inputs and reconstructions. However, since we will use the same matrix D to decode all of the points, we can no longer consider the points in isolation. Instead, we must minimize the Frobenius norm of the matrix of errors computed over all dimensions and all points:

D ∗ = arg min D

i,j

2 (i) x j − r(x(i)) j subject to DD = Il

(2.68)

To derive the algorithm for ﬁnding D∗ , we will start by considering the case where l = 1. In this case, D is just a single vector, d . Substituting Eq. 2.67 into Eq. 2.68 and simplifying D into d, the problem reduces to d ∗ = arg min d

i

||x(i) − ddx(i)|| 22 subject to ||d||2 = 1.

(2.69)

The above formulation is the most direct way of performing the substitution, but is not the most stylistically pleasing way to write the equation. It places the scalar value d x (i) on the right of the vector d. It is more conventional to write scalar coeﬃcients on the left of vector they operate on. We therefore usually write such a formula as ||x(i) − dx (i)d||22 subject to ||d||2 = 1, d ∗ = arg min (2.70) d

i

or, exploiting the fact that a scalar is its own transpose, as d ∗ = arg min ||x(i) − x(i) dd||22 subject to ||d||2 = 1. d

(2.71)

i

The reader should aim to become familiar with such cosmetic rearrangements. At this point, it can be helpful to rewrite the problem in terms of a single design matrix of examples, rather than as a sum over separate example vectors. This will allow us to use more compact notation. Let X ∈ Rm×n be the matrix deﬁned by stacking all of the vectors describing the points, such that Xi,: = x (i) . We can now rewrite the problem as d∗ = arg min ||X − Xdd ||2F subject to d d = 1. d

50

(2.72)

CHAPTER 2. LINEAR ALGEBRA

Disregarding the constraint for the moment, we can simplify the Frobenius norm portion as follows: arg min ||X − Xdd ||2F (2.73) d

= arg min Tr d

(by Eq. 2.49)

X − Xdd

X − Xdd

(2.74)

= arg min Tr(X X − X Xdd − dd X X + dd X Xdd )

(2.75)

d

= arg min Tr(X X) − Tr(X Xdd ) − Tr(dd X X) + Tr(dd X Xdd ) d

= arg min − Tr(X Xdd ) − Tr(dd X X) + Tr(dd X Xdd )

(2.76) (2.77)

d

(because terms not involving d do not aﬀect the arg min) = arg min −2 Tr(X Xdd) + Tr(dd X Xdd )

(2.78)

d

(because we can cycle the order of the matrices inside a trace, Eq. 2.52) = arg min −2 Tr(X Xdd) + Tr(X Xdddd )

(2.79)

d

(using the same property again) At this point, we re-introduce the constraint: arg min −2 Tr(X Xdd ) + Tr(X Xdddd ) subject to d d = 1

(2.80)

= arg min −2 Tr(X Xdd ) + Tr(X Xdd ) subject to dd = 1

(2.81)

d

d

(due to the constraint) = arg min − Tr(X Xdd ) subject to dd = 1

(2.82)

= arg max Tr(X Xdd ) subject to d d = 1

(2.83)

d

d

= arg max Tr(d X Xd) subject to d d = 1 d

51

(2.84)

CHAPTER 2. LINEAR ALGEBRA

This optimization problem may be solved using eigendecomposition. Speciﬁcally, the optimal d is given by the eigenvector of X X corresponding to the largest eigenvalue. In the general case, where l > 1, the matrix D is given by the l eigenvectors corresponding to the largest eigenvalues. This may be shown using proof by induction. We recommend writing this proof as an exercise. Linear algebra is one of the fundamental mathematical disciplines that is necessary to understand deep learning. Another key area of mathematics that is ubiquitous in machine learning is probability theory, presented next.

52

Chapter 3

Probability and Information Theory In this chapter, we describe probability theory and information theory. Probability theory is a mathematical framework for representing uncertain statements. It provides a means of quantifying uncertainty and axioms for deriving new uncertain statements. In artiﬁcial intelligence applications, we use probability theory in two major ways. First, the laws of probability tell us how AI systems should reason, so we design our algorithms to compute or approximate various expressions derived using probability theory. Second, we can use probability and statistics to theoretically analyze the behavior of proposed AI systems. Probability theory is a fundamental tool of many disciplines of science and engineering. We provide this chapter to ensure that readers whose background is primarily in software engineering with limited exposure to probability theory can understand the material in this book. While probability theory allows us to make uncertain statements and reason in the presence of uncertainty, information allows us to quantify the amount of uncertainty in a probability distribution. If you are already familiar with probability theory and information theory, you may wish to skip all of this chapter except for Sec. 3.14, which describes the graphs we use to describe structured probabilistic models for machine learning. If you have absolutely no prior experience with these subjects, this chapter should be suﬃcient to successfully carry out deep learning research projects, but we do suggest that you consult an additional resource, such as Jaynes (2003).

53

CHAPTER 3. PROBABILITY AND INFORMATION THEORY

3.1

Why Probability?

Many branches of computer science deal mostly with entities that are entirely deterministic and certain. A programmer can usually safely assume that a CPU will execute each machine instruction ﬂawlessly. Errors in hardware do occur, but are rare enough that most software applications do not need to be designed to account for them. Given that many computer scientists and software engineers work in a relatively clean and certain environment, it can be surprising that machine learning makes heavy use of probability theory. This is because machine learning must always deal with uncertain quantities, and sometimes may also need to deal with stochastic (non-deterministic) quantities. Uncertainty and stochasticity can arise from many sources. Researchers have made compelling arguments for quantifying uncertainty using probability since at least the 1980s. Many of the arguments presented here are summarized from or inspired by Pearl (1988). Nearly all activities require some ability to reason in the presence of uncertainty. In fact, beyond mathematical statements that are true by deﬁnition, it is diﬃcult to think of any proposition that is absolutely true or any event that is absolutely guaranteed to occur. There are three possible sources of uncertainty: 1. Inherent stochasticity in the system being modeled. For example, most interpretations of quantum mechanics describe the dynamics of subatomic particles as being probabilistic. We can also create theoretical scenarios that we postulate to have random dynamics, such as a hypothetical card game where we assume that the cards are truly shuﬄed into a random order. 2. Incomplete observability. Even deterministic systems can appear stochastic when we cannot observe all of the variables that drive the behavior of the system. For example, in the Monty Hall problem, a game show contestant is asked to choose between three doors and wins a prize held behind the chosen door. Two doors lead to a goat while a third leads to a car. The outcome given the contestant’s choice is deterministic, but from the contestant’s point of view, the outcome is uncertain. 3. Incomplete modeling. When we use a model that must discard some of the information we have observed, the discarded information results in uncertainty in the model’s predictions. For example, suppose we build a robot that can exactly observe the location of every object around it. If the 54

CHAPTER 3. PROBABILITY AND INFORMATION THEORY

robot discretizes space when predicting the future location of these objects, then the discretization makes the robot immediately become uncertain about the precise position of objects: each object could be anywhere within the discrete cell that it was observed to occupy. In many cases, it is more practical to use a simple but uncertain rule rather than a complex but certain one, even if the true rule is deterministic and our modeling system has the ﬁdelity to accommodate a complex rule. For example, the simple rule “Most birds ﬂy” is cheap to develop and is broadly useful, while a rule of the form, “Birds ﬂy, except for very young birds that have not yet learned to ﬂy, sick or injured birds that have lost the ability to ﬂy, ﬂightless species of birds including the cassowary, ostrich and kiwi. . . ” is expensive to develop, maintain and communicate, and after all of this eﬀort is still very brittle and prone to failure. Given that we need a means of representing and reasoning about uncertainty, it is not immediately obvious that probability theory can provide all of the tools we want for artiﬁcial intelligence applications. Probability theory was originally developed to analyze the frequencies of events. It is easy to see how probability theory can be used to study events like drawing a certain hand of cards in a game of poker. These kinds of events are often repeatable. When we say that an outcome has a probability p of occurring, it means that if we repeated the experiment (e.g., draw a hand of cards) inﬁnitely many times, then proportion p of the repetitions would result in that outcome. This kind of reasoning does not seem immediately applicable to propositions that are not repeatable. If a doctor analyzes a patient and says that the patient has a 40% chance of having the ﬂu, this means something very diﬀerent—we can not make inﬁnitely many replicas of the patient, nor is there any reason to believe that diﬀerent replicas of the patient would present with the same symptoms yet have varying underlying conditions. In the case of the doctor diagnosing the patient, we use probability to represent a degree of belief, with 1 indicating absolute certainty that the patient has the ﬂu and 0 indicating absolute certainty that the patient does not have the ﬂu. The former kind of probability, related directly to the rates at which events occur, is known as frequentist probability, while the latter, related to qualitative levels of certainty, is known as Bayesian probability. If we list several properties that we expect common sense reasoning about uncertainty to have, then the only way to satisfy those properties is to treat Bayesian probabilities as behaving exactly the same as frequentist probabilities. For example, if we want to compute the probability that a player will win a poker game given that she has a certain set of cards, we use exactly the same formulas as when we compute the probability that a patient has a disease given that she 55

CHAPTER 3. PROBABILITY AND INFORMATION THEORY

has certain symptoms. For more details about why a small set of common sense assumptions implies that the same axioms must control both kinds of probability, see Ramsey (1926). Probability can be seen as the extension of logic to deal with uncertainty. Logic provides a set of formal rules for determining what propositions are implied to be true or false given the assumption that some other set of propositions is true or false. Probability theory provides a set of formal rules for determining the likelihood of a proposition being true given the likelihood of other propositions.

3.2

Random Variables

A random variable is a variable that can take on diﬀerent values randomly. We typically denote the random variable itself with a lower case letter in plain typeface, and the values it can take on with lower case script letters. For example, x1 and x2 are both possible values that the random variable x can take on. For vector-valued variables, we would write the random variable as x and one of its values as x. On its own, a random variable is just a description of the states that are possible; it must be coupled with a probability distribution that speciﬁes how likely each of these states are. Random variables may be discrete or continuous. A discrete random variable is one that has a ﬁnite or countably inﬁnite number of states. Note that these states are not necessarily the integers; they can also just be named states that are not considered to have any numerical value. A continuous random variable is associated with a real value.

3.3

Probability Distributions

A probability distribution is a description of how likely a random variable or set of random variables is to take on each of its possible states. The way we describe probability distributions depends on whether the variables are discrete or continuous.

3.3.1

Discrete Variables and Probability Mass Functions

A probability distribution over discrete variables may be described using a probability mass function (PMF). We typically denote probability mass functions with a capital P . Often we associate each random variable with a diﬀerent probability 56

CHAPTER 3. PROBABILITY AND INFORMATION THEORY

mass function and the reader must infer which probability mass function to use based on the identity of the random variable, rather than the name of the function; P (x) is usually not the same as P (y). The probability mass function maps from a state of a random variable to the probability of that random variable taking on that state. The probability that x = x is denoted as P (x), with a probability of 1 indicating that x = x is certain and a probability of 0 indicating that x = x is impossible. Sometimes to disambiguate which PMF to use, we write the name of the random variable explicitly: P (x = x). Sometimes we deﬁne a variable ﬁrst, then use ∼ notation to specify which distribution it follows later: x ∼ P (x). Probability mass functions can act on many variables at the same time. Such a probability distribution over many variables is known as a joint probability distribution. P (x = x, y = y) denotes the probability that x = x and y = y simultaneously. We may also write P (x, y) for brevity. To be a probability mass function on a random variable x, a function P must satisfy the following properties: • The domain of P must be the set of all possible states of x. • ∀x ∈ x,0 ≤ P (x) ≤ 1. An impossible event has probability 0 and no state can be less probable than that. Likewise, an event that is guaranteed to happen has probability 1, and no state can have a greater chance of occurring. • x∈x P (x) = 1. We refer to this property as being normalized. Without this property, we could obtain probabilities greater than one by computing the probability of one of many events occurring. For example, consider a single discrete random variable x with k diﬀerent states. We can place a uniform distribution on x—that is, make each of its states equally likely—by setting its probability mass function to P (x = x i ) =

1 k

(3.1)

for all i. We can see that this ﬁts the requirements for a probability mass function. The value 1k is positive because k is a positive integer. We also see that

P (x = xi ) =

i

1 i

so the distribution is properly normalized. 57

k

=

k = 1, k

(3.2)

CHAPTER 3. PROBABILITY AND INFORMATION THEORY

3.3.2

Continuous Variables and Probability Density Functions

When working with continuous random variables, we describe probability distributions using a probability density function (PDF) rather than a probability mass function. To be a probability density function, a function p must satisfy the following properties: • The domain of p must be the set of all possible states of x. • ∀x ∈ x, p(x) ≥ 0. Note that we do not require p(x) ≤ 1. • p(x)dx = 1.

A probability density function p(x) does not give the probability of a speciﬁc state directly, instead the probability of landing inside an inﬁnitesimal region with volume δx is given by p(x)δx. We can integrate the density function to ﬁnd the actual probability mass of a set of points. Speciﬁcally, the probability that x lies in some set S is given by the integral of p (x) over that set. In the univariate example, the probability that x lies in the interval [a, b] is given by [a,b] p(x)dx.

For an example of a probability density function corresponding to a speciﬁc probability density over a continuous random variable, consider a uniform distribution on an interval of the real numbers. We can do this with a function u(x; a, b), where a and b are the endpoints of the interval, with b > a. The “;” notation means “parametrized by”; we consider x to be the argument of the function, while a and b are parameters that deﬁne the function. To ensure that there is no probability mass outside the interval, we say u(x; a, b) = 0 for all x ∈[a, b]. Within [ a, b], u(x; a, b) = b−1 a . We can see that this is nonnegative everywhere. Additionally, it integrates to 1. We often denote that x follows the uniform distribution on [ a, b] by writing x ∼ U (a, b).

3.4

Marginal Probability

Sometimes we know the probability distribution over a set of variables and we want to know the probability distribution over just a subset of them. The probability distribution over the subset is known as the marginal probability distribution. For example, suppose we have discrete random variables x and y, and we know P (x, y). We can ﬁnd P (x) with the sum rule: ∀x ∈ x, P (x = x) = P (x = x, y = y ). (3.3) y

58

CHAPTER 3. PROBABILITY AND INFORMATION THEORY

The name “marginal probability” comes from the process of computing marginal probabilities on paper. When the values of P (x, y ) are written in a grid with diﬀerent values of x in rows and diﬀerent values of y in columns, it is natural to sum across a row of the grid, then write P(x) in the margin of the paper just to the right of the row. For continuous variables, we need to use integration instead of summation: p(x) = p(x, y )dy. (3.4)

3.5

Conditional Probability

In many cases, we are interested in the probability of some event, given that some other event has happened. This is called a conditional probability. We denote the conditional probability that y = y given x = x as P (y = y | x = x). This conditional probability can be computed with the formula P (y = y | x = x) =

P (y = y, x = x) . P (x = x)

(3.5)

The conditional probability is only deﬁned when P( x = x) > 0. We cannot compute the conditional probability conditioned on an event that never happens. It is important not to confuse conditional probability with computing what would happen if some action were undertaken. The conditional probability that a person is from Germany given that they speak German is quite high, but if a randomly selected person is taught to speak German, their country of origin does not change. Computing the consequences of an action is called making an intervention query. Intervention queries are the domain of causal modeling, which we do not explore in this book.

3.6

The Chain Rule of Conditional Probabilities

Any joint probability distribution over many random variables may be decomposed into conditional distributions over only one variable: P (x(1) , . . . , x(n) ) = P (x(1) )Πni=2 P (x(i) | x (1) , . . . , x(i−1) ).

(3.6)

This observation is known as the chain rule or product rule of probability. It follows immediately from the deﬁnition of conditional probability in Eq. 3.5. For 59

CHAPTER 3. PROBABILITY AND INFORMATION THEORY

example, applying the deﬁnition twice, we get P (a, b, c) = P (a | b, c)P (b, c) P (b, c) = P (b | c)P (c)

P (a, b, c) = P (a | b, c)P (b | c)P (c).

3.7

Independence and Conditional Independence

Two random variables x and y are independent if their probability distribution can be expressed as a product of two factors, one involving only x and one involving only y: ∀x ∈ x, y ∈ y, p(x = x, y = y ) = p(x = x)p(y = y).

(3.7)

Two random variables x and y are conditionally independent given a random variable z if the conditional probability distribution over x and y factorizes in this way for every value of z:

∀x ∈ x, y ∈ y, z ∈ z, p(x = x, y = y | z = z) = p(x = x | z = z )p(y = y | z = z). (3.8) We can denote independence and conditional independence with compact notation: x⊥y means that x and y are independent, while x⊥y | z means that x and y are conditionally independent given z.

3.8

Expectation, Variance and Covariance

The expectation or expected value of some function f (x ) with respect to a probability distribution P (x) is the average or mean value that f takes on when x is drawn from P . For discrete variables this can be computed with a summation: [ f ( x )] = P (x)f (x), (3.9) Ex∼P x

while for continuous variables, it is computed with an integral: Ex∼p [f (x)] = p(x)f (x)dx. 60

(3.10)

CHAPTER 3. PROBABILITY AND INFORMATION THEORY

When the identity of the distribution is clear from the context, we may simply write the name of the random variable that the expectation is over, as in E x[f (x)]. If it is clear which random variable the expectation is over, we may omit the subscript entirely, as in E[f (x)]. By default, we can assume that E[·] averages over the values of all the random variables inside the brackets. Likewise, when there is no ambiguity, we may omit the square brackets. Expectations are linear, for example, Ex[αf (x) + βg(x)] = αEx [f (x)] + βE x[g(x)],

(3.11)

when α and β are not dependent on x. The variance gives a measure of how much the values of a function of a random variable x vary as we sample diﬀerent values of x from its probability distribution: 2 Var(f (x)) = E (f (x) − E[f (x)]) . (3.12) When the variance is low, the values of f(x) cluster near their expected value. The square root of the variance is known as the standard deviation.

The covariance gives some sense of how much two values are linearly related to each other, as well as the scale of these variables: Cov(f (x), g(y)) = E [(f (x) − E [f (x)]) (g (y) − E [g(y)])] .

(3.13)

High absolute values of the covariance mean that the values change very much and are both far from their respective means at the same time. If the sign of the covariance is positive, then both variables tend to take on relatively high values simultaneously. If the sign of the covariance is negative, then one variable tends to take on a relatively high value at the times that the other takes on a relatively low value and vice versa. Other measures such as correlation normalize the contribution of each variable in order to measure only how much the variables are related, rather than also being aﬀected by the scale of the separate variables. The notions of covariance and dependence are related, but are in fact distinct concepts. They are related because two variables that are independent have zero covariance, and two variables that have non-zero covariance are dependent. However, independence is a distinct property from covariance. For two variables to have zero covariance, there must be no linear dependence between them. Independence is a stronger requirement than zero covariance, because independence also excludes nonlinear relationships. It is possible for two variables to be dependent but have zero covariance. For example, suppose we ﬁrst sample a real number x from a uniform distribution over the interval [−1, 1]. We next sample a random variable 61

CHAPTER 3. PROBABILITY AND INFORMATION THEORY

s. With probability 12 , we choose the value of s to be 1. Otherwise, we choose the value of s to be − 1. We can then generate a random variable y by assigning y = sx. Clearly, x and y are not independent, because x completely determines the magnitude of y. However, Cov(x, y) = 0. The covariance matrix of a random vector x ∈ Rn is an n × n matrix, such that Cov(x) i,j = Cov(xi, x j).

(3.14)

The diagonal elements of the covariance give the variance: Cov(xi , xi) = Var(xi ).

3.9

(3.15)

Common Probability Distributions

Several simple probability distributions are useful in many contexts in machine learning.

3.9.1

Bernoulli Distribution

The Bernoulli distribution is a distribution over a single binary random variable. It is controlled by a single parameter φ ∈ [0, 1], which gives the probability of the random variable being equal to 1. It has the following properties:

3.9.2

P (x = 1) = φ

(3.16)

P (x = 0) = 1 − φ

(3.17)

P (x = x) = φx (1 − φ)1−x

(3.18)

Ex[x] = φ

(3.19)

Var x(x) = φ(1 − φ)

(3.20)

Multinoulli Distribution

The multinoulli or categorical distribution is a distribution over a single discrete variable with k diﬀerent states, where k is ﬁnite.1 The multinoulli distribution is 1

“Multinoulli” is a term that was recently coined by Gustavo Lacerdo and popularized by Murphy (2012). The multinoulli distribution is a special case of the multinomial distribution. A multinomial distribution is the distribution over vectors in {0, . . . , n}k representing how many times each of the k categories is visited when n samples are drawn from a multinoulli distribution. Many texts use the term “multinomial” to refer to multinoulli distributions without clarifying that they refer only to the n = 1 case. 62

CHAPTER 3. PROBABILITY AND INFORMATION THEORY

parametrized by a vector p ∈ [0, 1] k−1 , where p i gives the probability of the i-th state. The ﬁnal, k-th state’s probability is given by 1 − 1 p. Note that we must constrain 1 p ≤ 1. Multinoulli distributions are often used to refer to distributions over categories of objects, so we do not usually assume that state 1 has numerical value 1, etc. For this reason, we do not usually need to compute the expectation or variance of multinoulli-distributed random variables. The Bernoulli and multinoulli distributions are suﬃcient to describe any distribution over their domain. This is because they model discrete variables for which it is feasible to simply enumerate all of the states. When dealing with continuous variables, there are uncountably many states, so any distribution described by a small number of parameters must impose strict limits on the distribution.

3.9.3

Gaussian Distribution

The most commonly used distribution over real numbers is the normal distribution, also known as the Gaussian distribution: N (x; µ, σ 2) =

1 1 exp − 2 (x − µ)2 . 2πσ2 2σ

(3.21)

See Fig. 3.1 for a plot of the density function. The two parameters µ ∈ R and σ ∈ (0, ∞) control the normal distribution. The parameter µ gives the coordinate of the central peak. This is also the mean of the distribution: E[x] = µ. The standard deviation of the distribution is given by σ, and the variance by σ 2. When we evaluate the PDF, we need to square and invert σ. When we need to frequently evaluate the PDF with diﬀerent parameter values, a more eﬃcient way of parametrizing the distribution is to use a parameter β ∈ (0, ∞) to control the precision or inverse variance of the distribution: N (x; µ, β

−1

)=

β 1 2 exp − β (x − µ) . 2π 2

(3.22)

Normal distributions are a sensible choice for many applications. In the absence of prior knowledge about what form a distribution over the real numbers should take, the normal distribution is a good default choice for two major reasons. First, many distributions we wish to model are truly close to being normal distributions. The central limit theorem shows that the sum of many independent random variables is approximately normally distributed. This means that in 63

CHAPTER 3. PROBABILITY AND INFORMATION THEORY

Figure 3.1: The normal distribution: The normal distribution N (x; µ, σ 2) exhibits a classic “bell curve” shape, with the x coordinate of its central peak given by µ, and the width of its peak controlled by σ. In this example, we depict the standard normal distribution, with µ = 0 and σ = 1.

practice, many complicated systems can be modeled successfully as normally distributed noise, even if the system can be decomposed into parts with more structured behavior. Second, out of all possible probability distributions with the same variance, the normal distribution encodes the maximum amount of uncertainty over the real numbers. We can thus think of the normal distribution as being the one that inserts the least amount of prior knowledge into a model. Fully developing and justifying this idea requires more mathematical tools, and is postponed to Sec. 19.4.2. The normal distribution generalizes to R n , in which case it is known as the multivariate normal distribution. It may be parametrized with a positive deﬁnite symmetric matrix Σ: 1 1 N (x; µ, Σ) = exp − (x − µ)Σ −1 (x − µ) . (3.23) (2π) ndet(Σ) 2 The parameter µ still gives the mean of the distribution, though now it is vector-valued. The parameter Σ gives the covariance matrix of the distribution. As in the univariate case, when we wish to evaluate the PDF several times for 64

CHAPTER 3. PROBABILITY AND INFORMATION THEORY

many diﬀerent values of the parameters, the covariance is not a computationally eﬃcient way to parametrize the distribution, since we need to invert Σ to evaluate the PDF. We can instead use a precision matrix β: −1

N (x ; µ, β

)=

det(β) 1 exp − (x − µ) β(x − µ) . (2π) n 2

(3.24)

We often ﬁx the covariance matrix to be a diagonal matrix. An even simpler version is the isotropic Gaussian distribution, whose covariance matrix is a scalar times the identity matrix.

3.9.4

Exponential and Laplace Distributions

In the context of deep learning, we often want to have a probability distribution with a sharp point at x = 0. To accomplish this, we can use the exponential distribution: p(x; λ) = λ1x≥0 exp (−λx) . (3.25)

The exponential distribution uses the indicator function 1 x≥0 to assign probability zero to all negative values of x. A closely related probability distribution that allows us to place a sharp peak of probability mass at an arbitrary point µ is the Laplace distribution 1 |x − µ| Laplace(x; µ, γ ) = exp − . (3.26) 2γ γ

3.9.5

The Dirac Distribution and Empirical Distribution

In some cases, we wish to specify that all of the mass in a probability distribution clusters around a single point. This can be accomplished by deﬁning a PDF using the Dirac delta function, δ(x): p(x) = δ(x − µ).

(3.27)

The Dirac delta function is deﬁned such that it is zero-valued everywhere except 0, yet integrates to 1. The Dirac delta function is not an ordinary function that associates each value x with a real-valued output, instead it is a diﬀerent kind of mathematical object called a generalized function that is deﬁned in terms of its properties when integrated. We can think of the Dirac delta function as being the limit point of a series of functions that put less and less mass on all points other than µ. 65

CHAPTER 3. PROBABILITY AND INFORMATION THEORY

By deﬁning p(x) to be δ shifted by −µ we obtain an inﬁnitely narrow and inﬁnitely high peak of probability mass where x = µ. A common use of the Dirac delta distribution is as a component of an empirical distribution, m 1 pˆ(x) = (3.28) δ(x − x(i)) m i=1

which puts probability mass m1 on each of the m points x(1) , . . . , x (m) forming a given data set or collection of samples. The Dirac delta distribution is only necessary to deﬁne the empirical distribution over continuous variables. For discrete variables, the situation is simpler: an empirical distribution can be conceptualized as a multinoulli distribution, with a probability associated to each possible input value that is simply equal to the empirical frequency of that value in the training set. We can view the empirical distribution formed from a dataset of training examples as specifying the distribution that we sample from when we train a model on this dataset. Another important perspective on the empirical distribution is that it is the probability density that maximizes the likelihood of the training data (see Sec. 5.5).

3.9.6

Mixtures of Distributions

It is also common to deﬁne probability distributions by combining other simpler probability distributions. One common way of combining distributions is to construct a mixture distribution. A mixture distribution is made up of several component distributions. On each trial, the choice of which component distribution generates the sample is determined by sampling a component identity from a multinoulli distribution: P (x ) = P (c = i)P (x | c = i) (3.29) i

where P (c) is the multinoulli distribution over component identities. We have already seen one example of a mixture distribution: the empirical distribution over real-valued variables is a mixture distribution with one Dirac component for each training example. The mixture model is one simple strategy for combining probability distributions to create a richer distribution. In Chapter 16, we explore the art of building complex probability distributions from simple ones in more detail. 66

CHAPTER 3. PROBABILITY AND INFORMATION THEORY

The mixture model allows us to brieﬂy glimpse a concept that will be of paramount importance later—the latent variable. A latent variable is a random variable that we cannot observe directly. The component identity variable c of the mixture model provides an example. Latent variables may be related to x through the joint distribution, in this case, P (x, c) = P (x | c )P (c ). The distribution P (c) over the latent variable and the distribution P (x | c) relating the latent variables to the visible variables determines the shape of the distribution P (x) even though it is possible to describe P (x) without reference to the latent variable. Latent variables are discussed further in Sec. 16.5. A very powerful and common type of mixture model is the Gaussian mixture model, in which the components p(x | c = i) are Gaussians. Each component has a separately parametrized mean µ (i) and covariance Σ(i). Some mixtures can have more constraints. For example, the covariances could be shared across components via the constraint Σ(i) = Σ∀i. As with a single Gaussian distribution, the mixture of Gaussians might constrain the covariance matrix for each component to be diagonal or isotropic. In addition to the means and covariances, the parameters of a Gaussian mixture specify the prior probability α i = P (c = i) given to each component i. The word “prior” indicates that it expresses the model’s beliefs about c before it has observed x. By comparison, P(c | x) is a posterior probability, because it is computed after observation of x. A Gaussian mixture model is a universal approximator of densities, in the sense that any smooth density can be approximated with any speciﬁc, non-zero amount of error by a Gaussian mixture model with enough components. Fig. 3.2 shows samples from a Gaussian mixture model.

3.10

Useful Properties of Common Functions

Certain functions arise often while working with probability distributions, especially the probability distributions used in deep learning models. One of these functions is the logistic sigmoid: σ (x ) =

1 . 1 + exp(−x)

(3.30)

The logistic sigmoid is commonly used to produce the φ parameter of a Bernoulli distribution because its range is (0,1), which lies within the valid range of values for the φ parameter. See Fig. 3.3 for a graph of the sigmoid function. The sigmoid 67

x2

CHAPTER 3. PROBABILITY AND INFORMATION THEORY

x1

Figure 3.2: Samples from a Gaussian mixture model. In this example, there are three components. From left to right, the ﬁrst component has an isotropic covariance matrix, meaning it has the same amount of variance in each direction. The second has a diagonal covariance matrix, meaning it can control the variance separately along each axis-aligned direction. This example has more variance along the x2 axis than along the x1 axis. The third component has a full-rank covariance matrix, allowing it to control the variance separately along an arbitrary basis of directions.

function saturates when its argument is very positive or very negative, meaning that the function becomes very ﬂat and insensitive to small changes in its input. Another commonly encountered function is the softplus function (Dugas et al., 2001): ζ (x) = log (1 + exp(x)) . (3.31) The softplus function can be useful for producing the β or σ parameter of a normal distribution because its range is (0, ∞ ). It also arises commonly when manipulating expressions involving sigmoids. The name of the softplus function comes from the fact that it is a smoothed or “softened” version of x+ = max(0, x).

(3.32)

See Fig. 3.4 for a graph of the softplus function. The following properties are all useful enough that you may wish to memorize them: σ (x ) =

exp(x) exp(x) + exp(0)

d σ(x) = σ(x)(1 − σ(x)) dx 68

(3.33) (3.34)

CHAPTER 3. PROBABILITY AND INFORMATION THEORY

Figure 3.3: The logistic sigmoid function.

Figure 3.4: The softplus function.

69

CHAPTER 3. PROBABILITY AND INFORMATION THEORY

1 − σ(x) = σ (−x)

(3.35)

log σ(x) = −ζ (−x)

(3.36)

d ζ (x ) = σ (x ) dx −1

∀x ∈ (0, 1), σ

(3.37)

x (x) = log 1−x

(3.38)

∀x > 0, ζ−1 (x) = log (exp(x) − 1) x ζ (x) = σ(y)dy

(3.39)

ζ (x) − ζ (−x) = x

(3.41)

(3.40)

−∞

The function σ −1 (x) is called the logit in statistics, but this term is more rarely used in machine learning. Eq. 3.41 provides extra justiﬁcation for the name “softplus.” The softplus function is intended as a smoothed version of the positive part function, x + = max{0, x}. The positive part function is the counterpart of the negative part function, x− = max{0, −x}. To obtain a smooth function that is analogous to the negative part, one can use ζ (−x). Just as x can be recovered from its positive part and negative part via the identity x + − x− = x, it is also possible to recover x using the same relationship between ζ (x) and ζ (−x), as shown in Eq. 3.41.

3.11

Bayes’ Rule

We often ﬁnd ourselves in a situation where we know P ( y | x) and need to know P (x | y ). Fortunately, if we also know P (x), we can compute the desired quantity using Bayes’ rule: P (x )P (y | x ) P (x | y ) = . (3.42) P (y ) Note that while P (y) appears in the formula, it is usually feasible to compute P (y) = x P (y | x)P (x), so we do not need to begin with knowledge of P (y).

Bayes’ rule is straightforward to derive from the deﬁnition of conditional probability, but it is useful to know the name of this formula since many texts refer to it by name. It is named after the Reverend Thomas Bayes, who ﬁrst discovered a special case of the formula. The general version presented here was independently discovered by Pierre-Simon Laplace. 70

CHAPTER 3. PROBABILITY AND INFORMATION THEORY

3.12

Technical Details of Continuous Variables

A proper formal understanding of continuous random variables and probability density functions requires developing probability theory in terms of a branch of mathematics known as measure theory. Measure theory is beyond the scope of this textbook, but we can brieﬂy sketch some of the issues that measure theory is employed to resolve. In Sec. 3.3.2, we saw that the probability of a continuous vector-valued x lying in some set S is given by the integral of p (x) over the set S. Some choices of set S can produce paradoxes. For example, it is possible to construct two sets S1 and S2 such that p (x ∈ S1) + p(x ∈ S2 ) > 1 but S1 ∩ S2 = ∅. These sets are generally constructed making very heavy use of the inﬁnite precision of real numbers, for example by making fractal-shaped sets or sets that are deﬁned by transforming the set of rational numbers. 2 One of the key contributions of measure theory is to provide a characterization of the set of sets that we can compute the probability of without encountering paradoxes. In this book, we only integrate over sets with relatively simple descriptions, so this aspect of measure theory never becomes a relevant concern. For our purposes, measure theory is more useful for describing theorems that apply to most points in R n but do not apply to some corner cases. Measure theory provides a rigorous way of describing that a set of points is negligibly small. Such a set is said to have “measure zero.” We do not formally deﬁne this concept in this textbook. However, it is useful to understand the intuition that a set of measure zero occupies no volume in the space we are measuring. For example, within R 2, a line has measure zero, while a ﬁlled polygon has positive measure. Likewise, an individual point has measure zero. Any union of countably many sets that each have measure zero also has measure zero (so the set of all the rational numbers has measure zero, for instance). Another useful term from measure theory is “almost everywhere.” A property that holds almost everywhere holds throughout all of space except for on a set of measure zero. Because the exceptions occupy a negligible amount of space, they can be safely ignored for many applications. Some important results in probability theory hold for all discrete values but only hold “almost everywhere” for continuous values. Another technical detail of continuous variables relates to handling continuous random variables that are deterministic functions of one another. Suppose we have two random variables, x and y, such that y = g(x), where g is an invertible, con2

The Banach-Tarski theorem provides a fun example of such sets. 71

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tinuous, diﬀerentiable transformation. One might expect that p y (y) = p x (g−1(y )). This is actually not the case. As a simple example, suppose we have scalar random variables x and y. Suppose y = x2 and x ∼ U(0, 1). If we use the rule p y(y) = p x(2y) then p y will be 0 everywhere except the interval [0, 12 ], and it will be 1 on this interval. This means 1 (3.43) p y (y)dy = , 2 which violates the deﬁnition of a probability distribution. This common mistake is wrong because it fails to account for the distortion of space introduced by the function g. Recall that the probability of x lying in an inﬁnitesimally small region with volume δx is given by p(x)δx. Since g can expand or contract space, the inﬁnitesimal volume surrounding x in x space may have diﬀerent volume in y space. To see how to correct the problem, we return to the scalar case. We need to preserve the property |py(g(x))dy| = |px (x)dx|. (3.44) Solving from this, we obtain p y(y) = px (g or equivalently

−1

∂x (y)) ∂y

∂g(x) . p x (x) = p y(g(x)) ∂x

(3.45)

(3.46)

In higher dimensions, the derivative generalizes to the determinant of the Jacobian i matrix—the matrix with J i,j = ∂x ∂y j . Thus, for real-valued vectors x and y ,

3.13

∂g( x ) . p x (x) = py(g(x)) det ∂x

(3.47)

Information Theory

Information theory is a branch of applied mathematics that revolves around quantifying how much information is present in a signal. It was originally invented to study sending messages from discrete alphabets over a noisy channel, such as communication via radio transmission. In this context, information theory tells how to design optimal codes and calculate the expected length of messages sampled from 72

CHAPTER 3. PROBABILITY AND INFORMATION THEORY

speciﬁc probability distributions using various encoding schemes. In the context of machine learning, we can also apply information theory to continuous variables where some of these message length interpretations do not apply. This ﬁeld is fundamental to many areas of electrical engineering and computer science. In this textbook, we mostly use a few key ideas from information theory to characterize probability distributions or quantify similarity between probability distributions. For more detail on information theory, see Cover and Thomas (2006) or MacKay (2003). The basic intuition behind information theory is that learning that an unlikely event has occurred is more informative than learning that a likely event has occurred. A message saying “the sun rose this morning” is so uninformative as to be unnecessary to send, but a message saying “there was a solar eclipse this morning” is very informative. We would like to quantify information in a way that formalizes this intuition. Speciﬁcally, • Likely events should have low information content, and in the extreme case, events that are guaranteed to happen should have no information content whatsoever. • Less likely events should have higher information content. • Independent events should have additive information. For example, ﬁnding out that a tossed coin has come up as heads twice should convey twice as much information as ﬁnding out that a tossed coin has come up as heads once. In order to satisfy all three of these properties, we deﬁne the self-information of an event x = x to be I (x) = − log P (x). (3.48) In this book, we always use log to mean the natural logarithm, with base e. Our deﬁnition of I(x) is therefore written in units of nats. One nat is the amount of information gained by observing an event of probability 1e . Other texts use base-2 logarithms and units called bits or shannons; information measured in bits is just a rescaling of information measured in nats. When x is continuous, we use the same deﬁnition of information by analogy, but some of the properties from the discrete case are lost. For example, an event with unit density still has zero information, despite not being an event that is guaranteed to occur. 73

CHAPTER 3. PROBABILITY AND INFORMATION THEORY

Figure 3.5: This plot shows how distributions that are closer to deterministic have low Shannon entropy while distributions that are close to uniform have high Shannon entropy. On the horizontal axis, we plot p , the probability of a binary random variable being equal to 1. The entropy is given by (p −1) log(1 − p) − p log p . When p is near 0, the distribution is nearly deterministic, because the random variable is nearly always 0. When p is near 1, the distribution is nearly deterministic, because the random variable is nearly always 1. When p = 0.5, the entropy is maximal, because the distribution is uniform over the two outcomes.

Self-information deals only with a single outcome. We can quantify the amount of uncertainty in an entire probability distribution using the Shannon entropy: H(x) = Ex∼P [I (x)] = −Ex∼P [log P (x)].

(3.49)

also denoted H(P ). In other words, the Shannon entropy of a distribution is the expected amount of information in an event drawn from that distribution. It gives a lower bound on the number of bits (if the logarithm is base 2, otherwise the units are diﬀerent) needed on average to encode symbols drawn from a distribution P. Distributions that are nearly deterministic (where the outcome is nearly certain) have low entropy; distributions that are closer to uniform have high entropy. See Fig. 3.5 for a demonstration. When x is continuous, the Shannon entropy is known as the diﬀerential entropy. If we have two separate probability distributions P ( x) and Q( x) over the same random variable x, we can measure how diﬀerent these two distributions are using the Kullback-Leibler (KL) divergence: P (x ) D KL (P Q) = Ex∼P log = E x∼P [log P (x) − log Q(x)] . (3.50) Q(x) 74

CHAPTER 3. PROBABILITY AND INFORMATION THEORY

In the case of discrete variables, it is the extra amount of information (measured in bits if we use the base 2 logarithm, but in machine learning we usually use nats and the natural logarithm) needed to send a message containing symbols drawn from probability distribution P , when we use a code that was designed to minimize the length of messages drawn from probability distribution Q. The KL divergence has many useful properties, most notably that it is nonnegative. The KL divergence is 0 if and only if P and Q are the same distribution in the case of discrete variables, or equal “almost everywhere” in the case of continuous variables. Because the KL divergence is non-negative and measures the diﬀerence between two distributions, it is often conceptualized as measuring some sort of distance between these distributions. However, it is not a true distance measure = DKL(QP ) for some P and Q. This because it is not symmetric: DKL(P Q) asymmetry means that there are important consequences to the choice of whether to use DKL(P Q) or D KL(QP ). See Fig. 3.6 for more detail. A quantity that is closely related to the KL divergence is the cross-entropy H (P, Q) = H (P ) + D KL(P Q ), which is similar to the KL divergence but lacking the term on the left: (3.51) H (P, Q) = −Ex∼P log Q(x). Minimizing the cross-entropy with respect to Q is equivalent to minimizing the KL divergence, because Q does not participate in the omitted term. When computing many of these quantities, it is common to encounter expressions of the form 0 log 0. By convention, in the context of information theory, we treat these expressions as limx→0 x log x = 0.

3.14

Structured Probabilistic Models

Machine learning algorithms often involve probability distributions over a very large number of random variables. Often, these probability distributions involve direct interactions between relatively few variables. Using a single function to describe the entire joint probability distribution can be very ineﬃcient (both computationally and statistically). Instead of using a single function to represent a probability distribution, we can split a probability distribution into many factors that we multiply together. For example, suppose we have three random variables: a, b and c. Suppose that a inﬂuences the value of b and b inﬂuences the value of c, but that a and c are independent given b. We can represent the probability distribution over all three 75

CHAPTER 3. PROBABILITY AND INFORMATION THEORY

q ∗ = argmin qD KL(pq )

q ∗ = argmin qDKL (q p) p(x) q ∗ (x)

Probability Density

Probability Density

p(x) q∗ (x)

x

x

Figure 3.6: The KL divergence is asymmetric. Suppose we have a distribution p(x ) and wish to approximate it with another distribution q(x). We have the choice of minimizing either DKL (pq ) or DKL (qp). We illustrate the eﬀect of this choice using a mixture of two Gaussians for p, and a single Gaussian for q. The choice of which direction of the KL divergence to use is problem-dependent. Some applications require an approximation that usually places high probability anywhere that the true distribution places high probability, while other applications require an approximation that rarely places high probability anywhere that the true distribution places low probability. The choice of the direction of the KL divergence reﬂects which of these considerations takes priority for each application. (Left) The eﬀect of minimizing D KL(pq). In this case, we select a q that has high probability where p has high probability. When p has multiple modes, q chooses to blur the modes together, in order to put high probability mass on all of them. (Right) The eﬀect of minimizing DKL (q p). In this case, we select a q that has low probability where p has low probability. When p has multiple modes that are suﬃciently widely separated, as in this ﬁgure, the KL divergence is minimized by choosing a single mode, in order to avoid putting probability mass in the low-probability areas between modes of p. Here, we illustrate the outcome when q is chosen to emphasize the left mode. We could also have achieved an equal value of the KL divergence by choosing the right mode. If the modes are not separated by a suﬃciently strong low probability region, then this direction of the KL divergence can still choose to blur the modes.

76

CHAPTER 3. PROBABILITY AND INFORMATION THEORY

variables as a product of probability distributions over two variables: p(a, b, c) = p(a)p(b | a)p(c | b).

(3.52)

These factorizations can greatly reduce the number of parameters needed to describe the distribution. Each factor uses a number of parameters that is exponential in the number of variables in the factor. This means that we can greatly reduce the cost of representing a distribution if we are able to ﬁnd a factorization into distributions over fewer variables. We can describe these kinds of factorizations using graphs. Here we use the word “graph” in the sense of graph theory: a set of vertices that may be connected to each other with edges. When we represent the factorization of a probability distribution with a graph, we call it a structured probabilistic model or graphical model. There are two main kinds of structured probabilistic models: directed and undirected. Both kinds of graphical models use a graph G in which each node in the graph corresponds to a random variable, and an edge connecting two random variables means that the probability distribution is able to represent direct interactions between those two random variables. Directed models use graphs with directed edges, and they represent factorizations into conditional probability distributions, as in the example above. Speciﬁcally, a directed model contains one factor for every random variable xi in the distribution, and that factor consists of the conditional distribution over xi given the parents of x i, denoted P a G(xi ): p(x) = p (xi | P a G (xi)) . (3.53) i

See Fig. 3.7 for an example of a directed graph and the factorization of probability distributions it represents.

Undirected models use graphs with undirected edges, and they represent factorizations into a set of functions; unlike in the directed case, these functions are usually not probability distributions of any kind. Any set of nodes that are all connected to each other in G is called a clique. Each clique C (i) in an undirected model is associated with a factor φ (i)(C(i) ). These factors are just functions, not probability distributions. The output of each factor must be non-negative, but there is no constraint that the factor must sum or integrate to 1 like a probability distribution. The probability of a conﬁguration of random variables is proportional to the product of all of these factors—assignments that result in larger factor values are 77

CHAPTER 3. PROBABILITY AND INFORMATION THEORY

a

b

c

d

e

Figure 3.7: A directed graphical model over random variables a, b, c, d and e. This graph corresponds to probability distributions that can be factored as p(a, b, c, d, e) = p(a)p(b | a)p(c | a, b)p(d | b)p(e | c).

(3.54)

This graph allows us to quickly see some properties of the distribution. For example, a and c interact directly, but a and e interact only indirectly via c.

more likely. Of course, there is no guarantee that this product will sum to 1. We therefore divide by a normalizing constant Z, deﬁned to be the sum or integral over all states of the product of the φ functions, in order to obtain a normalized probability distribution: 1 (i) (i) p(x) = φ . (3.55) C Z i

See Fig. 3.8 for an example of an undirected graph and the factorization of probability distributions it represents. Keep in mind that these graphical representations of factorizations are a language for describing probability distributions. They are not mutually exclusive families of probability distributions. Being directed or undirected is not a property of a probability distribution; it is a property of a particular description of a probability distribution, but any probability distribution may be described in both ways. Throughout Part I and Part II of this book, we will use structured probabilistic models merely as a language to describe which direct probabilistic relationships diﬀerent machine learning algorithms choose to represent. No further understanding of structured probabilistic models is needed until the discussion of research topics, in Part III, where we will explore structured probabilistic models in much greater detail. 78

CHAPTER 3. PROBABILITY AND INFORMATION THEORY

a

b

c

d

e

Figure 3.8: An undirected graphical model over random variables a, b, c, d and e. This graph corresponds to probability distributions that can be factored as p(a, b, c, d, e) =

1 (1) φ (a, b, c)φ(2) (b, d)φ(3) (c, e). Z

(3.56)

This graph allows us to quickly see some properties of the distribution. For example, a and c interact directly, but a and e interact only indirectly via c.

This chapter has reviewed the basic concepts of probability theory that are most relevant to deep learning. One more set of fundamental mathematical tools remains: numerical methods.

79

Chapter 4

Numerical Computation Machine learning algorithms usually require a high amount of numerical computation. This typically refers to algorithms that solve mathematical problems by methods that update estimates of the solution via an iterative process, rather than analytically deriving a formula providing a symbolic expression for the correct solution. Common operations include optimization (ﬁnding the value of an argument that minimizes or maximizes a function) and solving systems of linear equations. Even just evaluating a mathematical function on a digital computer can be diﬃcult when the function involves real numbers, which cannot be represented precisely using a ﬁnite amount of memory.

4.1

Overﬂow and Underﬂow

The fundamental diﬃculty in performing continuous math on a digital computer is that we need to represent inﬁnitely many real numbers with a ﬁnite number of bit patterns. This means that for almost all real numbers, we incur some approximation error when we represent the number in the computer. In many cases, this is just rounding error. Rounding error is problematic, especially when it compounds across many operations, and can cause algorithms that work in theory to fail in practice if they are not designed to minimize the accumulation of rounding error. One form of rounding error that is particularly devastating is underﬂow. Underﬂow occurs when numbers near zero are rounded to zero. Many functions behave qualitatively diﬀerently when their argument is zero rather than a small positive number. For example, we usually want to avoid division by zero (some software 80

CHAPTER 4. NUMERICAL COMPUTATION

environments will raise exceptions when this occurs, others will return a result with a placeholder not-a-number value) or taking the logarithm of zero (this is usually treated as −∞, which then becomes not-a-number if it is used for many further arithmetic operations). Another highly damaging form of numerical error is overﬂow. Overﬂow occurs when numbers with large magnitude are approximated as ∞ or −∞. Further arithmetic will usually change these inﬁnite values into not-a-number values. One example of a function that must be stabilized against underﬂow and overﬂow is the softmax function. The softmax function is often used to predict the probabilities associated with a multinoulli distribution. The softmax function is deﬁned to be exp(xi ) softmax(x)i = n . (4.1) exp(x ) j j=1

Consider what happens when all of the xi are equal to some constant c. Analytically, we can see that all of the outputs should be equal to 1n . Numerically, this may not occur when c has large magnitude. If c is very negative, then exp(c) will underﬂow. This means the denominator of the softmax will become 0, so the ﬁnal result is undeﬁned. When c is very large and positive, exp(c) will overﬂow, again resulting in the expression as a whole being undeﬁned. Both of these diﬃculties can be resolved by instead evaluating softmax(z ) where z = x − max i xi. Simple algebra shows that the value of the softmax function is not changed analytically by adding or subtracting a scalar from the input vector. Subtracting maxi xi results in the largest argument to exp being 0, which rules out the possibility of overﬂow. Likewise, at least one term in the denominator has a value of 1, which rules out the possibility of underﬂow in the denominator leading to a division by zero. There is still one small problem. Underﬂow in the numerator can still cause the expression as a whole to evaluate to zero. This means that if we implement log softmax(x) by ﬁrst running the softmax subroutine then passing the result to the log function, we could erroneously obtain −∞. Instead, we must implement a separate function that calculates log softmax in a numerically stable way. The log softmax function can be stabilized using the same trick as we used to stabilize the softmax function. For the most part, we do not explicitly detail all of the numerical considerations involved in implementing the various algorithms described in this book. Developers of low-level libraries should keep numerical issues in mind when implementing deep learning algorithms. Most readers of this book can simply rely on lowlevel libraries that provide stable implementations. In some cases, it is possible to implement a new algorithm and have the new implementation automatically 81

CHAPTER 4. NUMERICAL COMPUTATION

stabilized. Theano (Bergstra et al., 2010; Bastien et al., 2012) is an example of a software package that automatically detects and stabilizes many common numerically unstable expressions that arise in the context of deep learning.

4.2

Poor Conditioning

Conditioning refers to how rapidly a function changes with respect to small changes in its inputs. Functions that change rapidly when their inputs are perturbed slightly can be problematic for scientiﬁc computation because rounding errors in the inputs can result in large changes in the output. Consider the function f(x) = A−1x. When A ∈ R n×n has an eigenvalue decomposition, its condition number is λi max . (4.2) i,j λj

This is the ratio of the magnitude of the largest and smallest eigenvalue. When this number is large, matrix inversion is particularly sensitive to error in the input. This sensitivity is an intrinsic property of the matrix itself, not the result of rounding error during matrix inversion. Poorly conditioned matrices amplify pre-existing errors when we multiply by the true matrix inverse. In practice, the error will be compounded further by numerical errors in the inversion process itself.

4.3

Gradient-Based Optimization

Most deep learning algorithms involve optimization of some sort. Optimization refers to the task of either minimizing or maximizing some function f (x) by altering x. We usually phrase most optimization problems in terms of minimizing f (x). Maximization may be accomplished via a minimization algorithm by minimizing −f (x).

The function we want to minimize or maximize is called the objective function or criterion. When we are minimizing it, we may also call it the cost function, loss function, or error function. In this book, we use these terms interchangeably, though some machine learning publications assign special meaning to some of these terms. We often denote the value that minimizes or maximizes a function with a superscript ∗. For example, we might say x∗ = arg min f (x). 82

CHAPTER 4. NUMERICAL COMPUTATION

Figure 4.1: An illustration of how the derivatives of a function can be used to follow the function downhill to a minimum. This technique is called gradient descent.

We assume the reader is already familiar with calculus, but provide a brief review of how calculus concepts relate to optimization here. Suppose we have a function y = f (x), where both x and y are real numbers. The derivative of this function is denoted as f (x) or as dy dx . The derivative f (x) gives the slope of f (x) at the point x. In other words, it speciﬁes how to scale a small change in the input in order to obtain the corresponding change in the output: f (x + ) ≈ f (x) + f (x). The derivative is therefore useful for minimizing a function because it tells us how to change x in order to make a small improvement in y. For example, we know that f (x − sign(f (x))) is less than f(x) for small enough . We can thus reduce f (x) by moving x in small steps with opposite sign of the derivative. This technique is called gradient descent (Cauchy, 1847). See Fig. 4.1 for an example of this technique.

When f (x) = 0, the derivative provides no information about which direction to move. Points where f (x) = 0 are known as critical points or stationary points. A local minimum is a point where f(x) is lower than at all neighboring points, so it is no longer possible to decrease f (x) by making inﬁnitesimal steps. A local maximum is a point where f(x) is higher than at all neighboring points, so it is 83

CHAPTER 4. NUMERICAL COMPUTATION

Figure 4.2: Examples of each of the three types of critical points in 1-D. A critical point is a point with zero slope. Such a point can either be a local minimum, which is lower than the neighboring points, a local maximum, which is higher than the neighboring points, or a saddle point, which has neighbors that are both higher and lower than the point itself.

not possible to increase f(x) by making inﬁnitesimal steps. Some critical points are neither maxima nor minima. These are known as saddle points. See Fig. 4.2 for examples of each type of critical point. A point that obtains the absolute lowest value of f (x) is a global minimum. It is possible for there to be only one global minimum or multiple global minima of the function. It is also possible for there to be local minima that are not globally optimal. In the context of deep learning, we optimize functions that may have many local minima that are not optimal, and many saddle points surrounded by very ﬂat regions. All of this makes optimization very diﬃcult, especially when the input to the function is multidimensional. We therefore usually settle for ﬁnding a value of f that is very low, but not necessarily minimal in any formal sense. See Fig. 4.3 for an example. We often minimize functions that have multiple inputs: f : R n → R. For the concept of “minimization” to make sense, there must still be only one (scalar) output. For functions with multiple inputs, we must make use of the concept of partial derivatives. The partial derivative ∂x∂ i f(x) measures how f changes as only the variable xi increases at point x. The gradient generalizes the notion of derivative to the case where the derivative is with respect to a vector: the gradient of f is the vector containing all of the partial derivatives, denoted ∇x f (x). Element i of the gradient is the partial derivative of f with respect to x i. In multiple dimensions, 84

CHAPTER 4. NUMERICAL COMPUTATION

Figure 4.3: Optimization algorithms may fail to ﬁnd a global minimum when there are multiple local minima or plateaus present. In the context of deep learning, we generally accept such solutions even though they are not truly minimal, so long as they correspond to signiﬁcantly low values of the cost function.

critical points are points where every element of the gradient is equal to zero. The directional derivative in direction u (a unit vector) is the slope of the function f in direction u. In other words, the directional derivative is the derivative of the function f (x + αu) with respect to α , evaluated at α = 0. Using the chain ∂ f (x + αu) = u ∇xf (x). rule, we can see that ∂α To minimize f , we would like to ﬁnd the direction in which f decreases the fastest. We can do this using the directional derivative: min

u,u u=1

=

min

u,u u=1

u∇ xf (x)

||u||2||∇ x f (x)||2 cos θ

(4.3) (4.4)

where θ is the angle between u and the gradient. Substituting in ||u||2 = 1 and ignoring factors that do not depend on u, this simpliﬁes to minu cos θ. This is minimized when u points in the opposite direction as the gradient. In other words, the gradient points directly uphill, and the negative gradient points directly downhill. We can decrease f by moving in the direction of the negative gradient. This is known as the method of steepest descent or gradient descent. Steepest descent proposes a new point x = x − ∇x f (x) 85

(4.5)

CHAPTER 4. NUMERICAL COMPUTATION

where is the learning rate, a positive scalar determining the size of the step. We can choose in several diﬀerent ways. A popular approach is to set to a small constant. Sometimes, we can solve for the step size that makes the directional derivative vanish. Another approach is to evaluate f (x − ∇x f (x)) for several values of and choose the one that results in the smallest objective function value. This last strategy is called a line search. Steepest descent converges when every element of the gradient is zero (or, in practice, very close to zero). In some cases, we may be able to avoid running this iterative algorithm, and just jump directly to the critical point by solving the equation ∇xf (x) = 0 for x.

Although gradient descent is limited to optimization in continuous spaces, the general concept of making small moves (that are approximately the best small move) towards better conﬁgurations can be generalized to discrete spaces. Ascending an objective function of discrete parameters is called hill climbing (Russel and Norvig, 2003).

4.3.1

Beyond the Gradient: Jacobian and Hessian Matrices

Sometimes we need to ﬁnd all of the partial derivatives of a function whose input and output are both vectors. The matrix containing all such partial derivatives is known as a Jacobian matrix. Speciﬁcally, if we have a function f : Rm → Rn, then the Jacobian matrix J ∈ Rn×m of f is deﬁned such that Ji,j = ∂x∂ j f (x)i.

We are also sometimes interested in a derivative of a derivative. This is known as a second derivative. For example, for a function f : Rn → R, the derivative 2 with respect to xi of the derivative of f with respect to x j is denoted as ∂x∂∂x f . i

d2 dx2

j

f by f (x). The second derivative tells In a single dimension, we can denote us how the ﬁrst derivative will change as we vary the input. This is important because it tells us whether a gradient step will cause as much of an improvement as we would expect based on the gradient alone. We can think of the second derivative as measuring curvature. Suppose we have a quadratic function (many functions that arise in practice are not quadratic but can be approximated well as quadratic, at least locally). If such a function has a second derivative of zero, then there is no curvature. It is a perfectly ﬂat line, and its value can be predicted using only the gradient. If the gradient is 1, then we can make a step of size along the negative gradient, and the cost function will decrease by . If the second derivative is negative, the function curves downward, so the cost function will actually decrease by more than . Finally, if the second derivative is positive, the function curves upward, so the cost function can decrease by less than . See Fig. 86

CHAPTER 4. NUMERICAL COMPUTATION

Positive curvature

f (x)

No curvature

f (x)

f (x)

Negative curvature

x

x

x

Figure 4.4: The second derivative determines the curvature of a function. Here we show quadratic functions with various curvature. The dashed line indicates the value of the cost function we would expect based on the gradient information alone as we make a gradient step downhill. In the case of negative curvature, the cost function actually decreases faster than the gradient predicts. In the case of no curvature, the gradient predicts the decrease correctly. In the case of positive curvature, the function decreases slower than expected and eventually begins to increase, so too large of step sizes can actually increase the function inadvertently.

4.4 to see how diﬀerent forms of curvature aﬀect the relationship between the value of the cost function predicted by the gradient and the true value. When our function has multiple input dimensions, there are many second derivatives. These derivatives can be collected together into a matrix called the Hessian matrix. The Hessian matrix H (f )(x) is deﬁned such that H (f )(x)i,j =

∂2 f (x ). ∂xi ∂xj

(4.6)

Equivalently, the Hessian is the Jacobian of the gradient. Anywhere that the second partial derivatives are continuous, the diﬀerential operators are commutative, i.e. their order can be swapped: ∂2 ∂2 f (x) = f (x ). ∂x i∂x j ∂xj ∂x i

(4.7)

This implies that Hi,j = H j,i , so the Hessian matrix is symmetric at such points. Most of the functions we encounter in the context of deep learning have a symmetric Hessian almost everywhere. Because the Hessian matrix is real and symmetric, we can decompose it into a set of real eigenvalues and an orthogonal basis of 87

CHAPTER 4. NUMERICAL COMPUTATION

eigenvectors. The second derivative in a speciﬁc direction represented by a unit vector d is given by d Hd. When d is an eigenvector of H , the second derivative in that direction is given by the corresponding eigenvalue. For other directions of d, the directional second derivative is a weighted average of all of the eigenvalues, with weights between 0 and 1, and eigenvectors that have smaller angle with d receiving more weight. The maximum eigenvalue determines the maximum second derivative and the minimum eigenvalue determines the minimum second derivative. The (directional) second derivative tells us how well we can expect a gradient descent step to perform. We can make a second-order Taylor series approximation to the function f (x) around the current point x(0): 1 f (x) ≈ f (x(0) ) + (x − x (0) ) g + (x − x(0) )H (x − x (0) ). 2

(4.8)

where g is the gradient and H is the Hessian at x(0). If we use a learning rate of , then the new point x will be given by x (0) − g. Substituting this into our approximation, we obtain 1 f (x (0) − g) ≈ f (x(0) ) − g g + 2 gHg. 2

(4.9)

There are three terms here: the original value of the function, the expected improvement due to the slope of the function, and the correction we must apply to account for the curvature of the function. When this last term is too large, the gradient descent step can actually move uphill. When gHg is zero or negative, the Taylor series approximation predicts that increasing forever will decrease f forever. In practice, the Taylor series is unlikely to remain accurate for large , so one must resort to more heuristic choices of in this case. When gHg is positive, solving for the optimal step size that decreases the Taylor series approximation of the function the most yields g g ∗ = . (4.10) g Hg In the worst case, when g aligns with the eigenvector of H corresponding to the 1 maximal eigenvalue λmax , then this optimal step size is given by λ max . To the extent that the function we minimize can be approximated well by a quadratic function, the eigenvalues of the Hessian thus determine the scale of the learning rate. The second derivative can be used to determine whether a critical point is a local maximum, a local minimum, or saddle point. Recall that on a critical point, f (x) = 0. When f (x) > 0, this means that f (x) increases as we move to the right, and f (x ) decreases as we move to the left. This means f (x − ) < 0 and 88

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f (x + ) > 0 for small enough . In other words, as we move right, the slope begins to point uphill to the right, and as we move left, the slope begins to point uphill to the left. Thus, when f (x) = 0 and f (x) > 0, we can conclude that x is a local minimum. Similarly, when f (x) = 0 and f (x) < 0, we can conclude that x is a local maximum. This is known as the second derivative test. Unfortunately, when f (x) = 0, the test is inconclusive. In this case x may be a saddle point, or a part of a ﬂat region. In multiple dimensions, we need to examine all of the second derivatives of the function. Using the eigendecomposition of the Hessian matrix, we can generalize the second derivative test to multiple dimensions. At a critical point, where ∇x f(x) = 0, we can examine the eigenvalues of the Hessian to determine whether the critical point is a local maximum, local minimum, or saddle point. When the Hessian is positive deﬁnite (all its eigenvalues are positive), the point is a local minimum. This can be seen by observing that the directional second derivative in any direction must be positive, and making reference to the univariate second derivative test. Likewise, when the Hessian is negative deﬁnite (all its eigenvalues are negative), the point is a local maximum. In multiple dimensions, it is actually possible to ﬁnd positive evidence of saddle points in some cases. When at least one eigenvalue is positive and at least one eigenvalue is negative, we know that x is a local maximum on one cross section of f but a local minimum on another cross section. See Fig. 4.5 for an example. Finally, the multidimensional second derivative test can be inconclusive, just like the univariate version. The test is inconclusive whenever all of the non-zero eigenvalues have the same sign, but at least one eigenvalue is zero. This is because the univariate second derivative test is inconclusive in the cross section corresponding to the zero eigenvalue. In multiple dimensions, there can be a wide variety of diﬀerent second derivatives at a single point, because there is a diﬀerent second derivative for each direction. The condition number of the Hessian measures how much the second derivatives vary. When the Hessian has a poor condition number, gradient descent performs poorly. This is because in one direction, the derivative increases rapidly, while in another direction, it increases slowly. Gradient descent is unaware of this change in the derivative so it does not know that it needs to explore preferentially in the direction where the derivative remains negative for longer. It also makes it diﬃcult to choose a good step size. The step size must be small enough to avoid overshooting the minimum and going uphill in directions with strong positive curvature. This usually means that the step size is too small to make signiﬁcant progress in other directions with less curvature. See Fig. 4.6 for an example. This issue can be resolved by using information from the Hessian matrix to 89

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Figure 4.5: A saddle point containing both positive and negative curvature. The function in this example is f (x) = x21 − x 22. Along the axis corresponding to x1, the function curves upward. This axis is an eigenvector of the Hessian and has a positive eigenvalue. Along the axis corresponding to x 2, the function curves downward. This direction is an eigenvector of the Hessian with negative eigenvalue. The name “saddle point” derives from the saddle-like shape of this function. This is the quintessential example of a function with a saddle point. In more than one dimension, it is not necessary to have an eigenvalue of 0 in order to get a saddle point: it is only necessary to have both positive and negative eigenvalues. We can think of a saddle point with both signs of eigenvalues as being a local maximum within one cross section and a local minimum within another cross section.

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20

x2

10 0 −10 −20 −30 −30 −20 −10

0 x1

10

20

Figure 4.6: Gradient descent fails to exploit the curvature information contained in the Hessian matrix. Here we use gradient descent to minimize a quadratic function f( x) whose Hessian matrix has condition number 5. This means that the direction of most curvature has ﬁve times more curvature than the direction of least curvature. In this case, the most curvature is in the direction [1,1] and the least curvature is in the direction [1, −1] . The red lines indicate the path followed by gradient descent. This very elongated quadratic function resembles a long canyon. Gradient descent wastes time repeatedly descending canyon walls, because they are the steepest feature. Because the step size is somewhat too large, it has a tendency to overshoot the bottom of the function and thus needs to descend the opposite canyon wall on the next iteration. The large positive eigenvalue of the Hessian corresponding to the eigenvector pointed in this direction indicates that this directional derivative is rapidly increasing, so an optimization algorithm based on the Hessian could predict that the steepest direction is not actually a promising search direction in this context.

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guide the search. The simplest method for doing so is known as Newton’s method. Newton’s method is based on using a second-order Taylor series expansion to approximate f (x) near some point x(0) : 1 f (x) ≈ f (x(0) )+(x−x(0) )∇ x f (x(0) )+ (x−x(0) ) H (f )(x(0) )(x−x(0)). (4.11) 2 If we then solve for the critical point of this function, we obtain: x∗ = x(0) − H (f )(x(0) )−1 ∇x f (x(0)).

(4.12)

When f is a positive deﬁnite quadratic function, Newton’s method consists of applying Eq. 4.12 once to jump to the minimum of the function directly. When f is not truly quadratic but can be locally approximated as a positive deﬁnite quadratic, Newton’s method consists of applying Eq. 4.12 multiple times. Iteratively updating the approximation and jumping to the minimum of the approximation can reach the critical point much faster than gradient descent would. This is a useful property near a local minimum, but it can be a harmful property near a saddle point. As discussed in Sec. 8.2.3, Newton’s method is only appropriate when the nearby critical point is a minimum (all the eigenvalues of the Hessian are positive), whereas gradient descent is not attracted to saddle points unless the gradient points toward them. Optimization algorithms such as gradient descent that use only the gradient are called ﬁrst-order optimization algorithms. Optimization algorithms such as Newton’s method that also use the Hessian matrix are called second-order optimization algorithms (Nocedal and Wright, 2006). The optimization algorithms employed in most contexts in this book are applicable to a wide variety of functions, but come with almost no guarantees. This is because the family of functions used in deep learning is quite complicated. In many other ﬁelds, the dominant approach to optimization is to design optimization algorithms for a limited family of functions. In the context of deep learning, we sometimes gain some guarantees by restricting ourselves to functions that are either Lipschitz continuous or have Lipschitz continuous derivatives. A Lipschitz continuous function is a function f whose rate of change is bounded by a Lipschitz constant L: ∀x, ∀y, |f (x) − f (y)| ≤ L||x − y ||2 .

(4.13)

This property is useful because it allows us to quantify our assumption that a small change in the input made by an algorithm such as gradient descent will have a small change in the output. Lipschitz continuity is also a fairly weak constraint, 92

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and many optimization problems in deep learning can be made Lipschitz continuous with relatively minor modiﬁcations. Perhaps the most successful ﬁeld of specialized optimization is convex optimization. Convex optimization algorithms are able to provide many more guarantees by making stronger restrictions. Convex optimization algorithms are applicable only to convex functions—functions for which the Hessian is positive semideﬁnite everywhere. Such functions are well-behaved because they lack saddle points and all of their local minima are necessarily global minima. However, most problems in deep learning are diﬃcult to express in terms of convex optimization. Convex optimization is used only as a subroutine of some deep learning algorithms. Ideas from the analysis of convex optimization algorithms can be useful for proving the convergence of deep learning algorithms. However, in general, the importance of convex optimization is greatly diminished in the context of deep learning. For more information about convex optimization, see Boyd and Vandenberghe (2004) or Rockafellar (1997).

4.4

Constrained Optimization

Sometimes we wish not only to maximize or minimize a function f(x) over all possible values of x. Instead we may wish to ﬁnd the maximal or minimal value of f (x) for values of x in some set S. This is known as constrained optimization. Points x that lie within the set S are called feasible points in constrained optimization terminology. We often wish to ﬁnd a solution that is small in some sense. A common approach in such situations is to impose a norm constraint, such as ||x|| ≤ 1.

One simple approach to constrained optimization is simply to modify gradient descent taking the constraint into account. If we use a small constant step size , we can make gradient descent steps, then project the result back into S. If we use a line search, we can search only over step sizes that yield new x points that are feasible, or we can project each point on the line back into the constraint region. When possible, this method can be made more eﬃcient by projecting the gradient into the tangent space of the feasible region before taking the step or beginning the line search (Rosen, 1960). A more sophisticated approach is to design a diﬀerent, unconstrained optimization problem whose solution can be converted into a solution to the original, constrained optimization problem. For example, if we want to minimize f(x) for x ∈ R2 with x constrained to have exactly unit L2 norm, we can instead minimize 93

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g(θ ) = f ([cos θ, sin θ] ) with respect to θ, then return [cos θ, sin θ] as the solution to the original problem. This approach requires creativity; the transformation between optimization problems must be designed speciﬁcally for each case we encounter. The Karush–Kuhn–Tucker (KKT) approach1 provides a very general solution to constrained optimization. With the KKT approach, we introduce a new function called the generalized Lagrangian or generalized Lagrange function. To deﬁne the Lagrangian, we ﬁrst need to describe S in terms of equations and inequalities. We want a description of S in terms of m functions g(i) and n functions h(j) so that S = {x | ∀i, g(i)(x) = 0 and ∀j, h (j) (x) ≤ 0}. The equations involving g(i) are called the equality constraints and the inequalities involving h (j) are called inequality constraints. We introduce new variables λ i and α j for each constraint, these are called the KKT multipliers. The generalized Lagrangian is then deﬁned as (4.14) L(x, λ, α) = f (x) + λ i g(i)(x) + α jh (j) (x). i

j

We can now solve a constrained minimization problem using unconstrained optimization of the generalized Lagrangian. Observe that, so long as at least one feasible point exists and f (x) is not permitted to have value ∞, then min max max L(x, λ, α). x

α,α≥0

λ

(4.15)

has the same optimal objective function value and set of optimal points x as min f (x). x∈S

(4.16)

This follows because any time the constraints are satisﬁed, max max L(x, λ, α) = f (x), α,α≥0

λ

(4.17)

while any time a constraint is violated, max max L(x, λ, α) = ∞. λ

α,α≥0

(4.18)

These properties guarantee that no infeasible point will ever be optimal, and that the optimum within the feasible points is unchanged. 1

The KKT approach generalizes the method of Lagrange multipliers which allows equality constraints but not inequality constraints. 94

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To perform constrained maximization, we can construct the generalized Lagrange function of −f (x), which leads to this optimization problem: (i) −f ( x ) + min max max λ i g (x ) + αj h(j) (x). (4.19) x

λ

α,α≥0

i

j

We may also convert this to a problem with maximization in the outer loop: max min min f (x) + λig (i) (x) − αj h(j) (x). (4.20) x

λ α,α≥0

i

j

The sign of the term for the equality constraints does not matter; we may deﬁne it with addition or subtraction as we wish, because the optimization is free to choose any sign for each λi. The inequality constraints are particularly interesting. We say that a constraint is active if h (i)(x∗ ) = 0. If a constraint is not active, then the solution to the problem found using that constraint would remain at least a local solution if that constraint were removed. It is possible that an inactive constraint excludes other solutions. For example, a convex problem with an entire region of globally optimal points (a wide, ﬂat, region of equal cost) could have a subset of this region eliminated by constraints, or a non-convex problem could have better local stationary points excluded by a constraint that is inactive at convergence. However, the point found at convergence remains a stationary point whether or not the inactive constraints are included. Because an inactive h (i) has negative value, then the solution to min x maxλ maxα,α≥0 L(x, λ, α) will have α i = 0. We can thus observe that at the solution, αh(x) = 0. In other words, for all i, we know that at least one of the constraints αi ≥ 0 and h (i)(x) ≤ 0 must be active at the solution. To gain some intuition for this idea, we can say that either the solution is on the boundary imposed by the inequality and we must use its KKT multiplier to inﬂuence the solution to x, or the inequality has no inﬂuence on the solution and we represent this by zeroing out its KKT multiplier. h(i) (x )

The properties that the gradient of the generalized Lagrangian is zero, all constraints on both x and the KKT multipliers are satisﬁed, and α h (x) = 0 are called the Karush-Kuhn-Tucker (KKT) conditions (Karush, 1939; Kuhn and Tucker, 1951). Together, these properties describe the optimal points of constrained optimization problems. For more information about the KKT approach, see Nocedal and Wright (2006).

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4.5

Example: Linear Least Squares

Suppose we want to ﬁnd the value of x that minimizes f (x ) =

1 ||Ax − b||22. 2

(4.21)

There are specialized linear algebra algorithms that can solve this problem eﬃciently. However, we can also explore how to solve it using gradient-based optimization as a simple example of how these techniques work. First, we need to obtain the gradient: ∇ x f (x) = A (Ax − b) = AAx − Ab.

(4.22)

We can then follow this gradient downhill, taking small steps. See Algorithm 4.1 for details. Algorithm 4.1 An algorithm to minimize f(x) = using gradient descent.

1 2

||Ax − b||22 with respect to x

Set the step size () and tolerance (δ ) to small, positive numbers. while ||A Ax − A b||2 > δ do x ← x − A Ax − A b end while One can also solve this problem using Newton’s method. In this case, because the true function is quadratic, the quadratic approximation employed by Newton’s method is exact, and the algorithm converges to the global minimum in a single step. Now suppose we wish to minimize the same function, but subject to the constraint xx ≤ 1. To do so, we introduce the Lagrangian L(x, λ) = f (x) + λ x x − 1 . (4.23)

We can now solve the problem

min max L(x, λ). x λ,λ≥0

(4.24)

The smallest-norm solution to the unconstrained least squares problem may be found using the Moore-Penrose pseudoinverse: x = A+ b. If this point is feasible, then it is the solution to the constrained problem. Otherwise, we must ﬁnd a 96

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solution where the constraint is active. By diﬀerentiating the Lagrangian with respect to x, we obtain the equation AAx − Ab + 2λx = 0.

(4.25)

This tells us that the solution will take the form x = (A A + 2λI ) −1A b.

(4.26)

The magnitude of λ must be chosen such that the result obeys the constraint. We can ﬁnd this value by performing gradient ascent on λ. To do so, observe ∂ L(x, λ) = x x − 1. ∂λ

(4.27)

When the norm of x exceeds 1, this derivative is positive, so to follow the derivative uphill and increase the Lagrangian with respect to λ, we increase λ. Because the coeﬃcient on the x x penalty has increased, solving the linear equation for x will now yield a solution with smaller norm. The process of solving the linear equation and adjusting λ continues until x has the correct norm and the derivative on λ is 0. This concludes the mathematical preliminaries that we use to develop machine learning algorithms. We are now ready to build and analyze some full-ﬂedged learning systems.

97

Chapter 5

Machine Learning Basics Deep learning is a speciﬁc kind of machine learning. In order to understand deep learning well, one must have a solid understanding of the basic principles of machine learning. This chapter provides a brief course in the most important general principles that will be applied throughout the rest of the book. Novice readers or those who want a wider perspective are encouraged to consider machine learning textbooks with a more comprehensive coverage of the fundamentals, such as Murphy (2012) or Bishop (2006). If you are already familiar with machine learning basics, feel free to skip ahead to Sec. 5.11. That section covers some perspectives on traditional machine learning techniques that have strongly inﬂuenced the development of deep learning algorithms. We begin with a deﬁnition of what a learning algorithm is, and present an example: the linear regression algorithm. We then proceed to describe how the challenge of ﬁtting the training data diﬀers from the challenge of ﬁnding patterns that generalize to new data. Most machine learning algorithms have settings called hyperparameters that must be determined external to the learning algorithm itself; we discuss how to set these using additional data. Machine learning is essentially a form of applied statistics with increased emphasis on the use of computers to statistically estimate complicated functions and a decreased emphasis on proving conﬁdence intervals around these functions; we therefore present the two central approaches to statistics: frequentist estimators and Bayesian inference. Most machine learning algorithms can be divided into the categories of supervised learning and unsupervised learning; we describe these categories and give some examples of simple learning algorithms from each category. Most deep learning algorithms are based on an optimization algorithm called stochastic gradient descent. We describe how to combine various algorithm components such as an 98

CHAPTER 5. MACHINE LEARNING BASICS

optimization algorithm, a cost function, a model, and a dataset to build a machine learning algorithm. Finally, in Sec. 5.11, we describe some of the factors that have limited the ability of traditional machine learning to generalize. These challenges have motivated the development of deep learning algorithms that overcome these obstacles.

5.1

Learning Algorithms

A machine learning algorithm is an algorithm that is able to learn from data. But what do we mean by learning? Mitchell (1997) provides the deﬁnition “A computer program is said to learn from experience E with respect to some class of tasks T and performance measure P , if its performance at tasks in T , as measured by P , improves with experience E.” One can imagine a very wide variety of experiences E, tasks T , and performance measures P , and we do not make any attempt in this book to provide a formal deﬁnition of what may be used for each of these entities. Instead, the following sections provide intuitive descriptions and examples of the diﬀerent kinds of tasks, performance measures and experiences that can be used to construct machine learning algorithms.

5.1.1

The Task, T

Machine learning allows us to tackle tasks that are too diﬃcult to solve with ﬁxed programs written and designed by human beings. From a scientiﬁc and philosophical point of view, machine learning is interesting because developing our understanding of machine learning entails developing our understanding of the principles that underlie intelligence. In this relatively formal deﬁnition of the word “task,” the process of learning itself is not the task. Learning is our means of attaining the ability to perform the task. For example, if we want a robot to be able to walk, then walking is the task. We could program the robot to learn to walk, or we could attempt to directly write a program that speciﬁes how to walk manually. Machine learning tasks are usually described in terms of how the machine learning system should process an example. An example is a collection of features that have been quantitatively measured from some object or event that we want the machine learning system to process. We typically represent an example as a vector x ∈ Rn where each entry x i of the vector is another feature. For example, the features of an image are usually the values of the pixels in the image. 99

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Many kinds of tasks can be solved with machine learning. Some of the most common machine learning tasks include the following: • Classiﬁcation: In this type of task, the computer program is asked to specify which of k categories some input belongs to. To solve this task, the learning algorithm is usually asked to produce a function f : Rn → {1, . . . , k}. When y = f (x), the model assigns an input described by vector x to a category identiﬁed by numeric code y. There are other variants of the classiﬁcation task, for example, where f outputs a probability distribution over classes. An example of a classiﬁcation task is object recognition, where the input is an image (usually described as a set of pixel brightness values), and the output is a numeric code identifying the object in the image. For example, the Willow Garage PR2 robot is able to act as a waiter that can recognize diﬀerent kinds of drinks and deliver them to people on command (Goodfellow et al., 2010). Modern object recognition is best accomplished with deep learning (Krizhevsky et al., 2012; Ioﬀe and Szegedy, 2015). Object recognition is the same basic technology that allows computers to recognize faces (Taigman et al., 2014), which can be used to automatically tag people in photo collections and allow computers to interact more naturally with their users. • Classiﬁcation with missing inputs: Classiﬁcation becomes more challenging if the computer program is not guaranteed that every measurement in its input vector will always be provided. In order to solve the classiﬁcation task, the learning algorithm only has to deﬁne a single function mapping from a vector input to a categorical output. When some of the inputs may be missing, rather than providing a single classiﬁcation function, the learning algorithm must learn a set of functions. Each function corresponds to classifying x with a diﬀerent subset of its inputs missing. This kind of situation arises frequently in medical diagnosis, because many kinds of medical tests are expensive or invasive. One way to eﬃciently deﬁne such a large set of functions is to learn a probability distribution over all of the relevant variables, then solve the classiﬁcation task by marginalizing out the missing variables. With n input variables, we can now obtain all 2n diﬀerent classiﬁcation functions needed for each possible set of missing inputs, but we only need to learn a single function describing the joint probability distribution. See Goodfellow et al. (2013b) for an example of a deep probabilistic model applied to such a task in this way. Many of the other tasks described in this section can also be generalized to work with missing inputs; classiﬁcation with missing inputs is just one example of what machine learning can do. 100

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• Regression: In this type of task, the computer program is asked to predict a numerical value given some input. To solve this task, the learning algorithm is asked to output a function f : R n → R. This type of task is similar to classiﬁcation, except that the format of output is diﬀerent. An example of a regression task is the prediction of the expected claim amount that an insured person will make (used to set insurance premiums), or the prediction of future prices of securities. These kinds of predictions are also used for algorithmic trading. • Transcription: In this type of task, the machine learning system is asked to observe a relatively unstructured representation of some kind of data and transcribe it into discrete, textual form. For example, in optical character recognition, the computer program is shown a photograph containing an image of text and is asked to return this text in the form of a sequence of characters (e.g., in ASCII or Unicode format). Google Street View uses deep learning to process address numbers in this way (Goodfellow et al., 2014d). Another example is speech recognition, where the computer program is provided an audio waveform and emits a sequence of characters or word ID codes describing the words that were spoken in the audio recording. Deep learning is a crucial component of modern speech recognition systems used at major companies including Microsoft, IBM and Google (Hinton et al., 2012b). • Machine translation: In a machine translation task, the input already consists of a sequence of symbols in some language, and the computer program must convert this into a sequence of symbols in another language. This is commonly applied to natural languages, such as to translate from English to French. Deep learning has recently begun to have an important impact on this kind of task (Sutskever et al., 2014; Bahdanau et al., 2015). • Structured output: Structured output tasks involve any task where the output is a vector (or other data structure containing multiple values) with important relationships between the diﬀerent elements. This is a broad category, and subsumes the transcription and translation tasks described above, but also many other tasks. One example is parsing—mapping a natural language sentence into a tree that describes its grammatical structure and tagging nodes of the trees as being verbs, nouns, or adverbs, and so on. See Collobert (2011) for an example of deep learning applied to a parsing task. Another example is pixel-wise segmentation of images, where the computer program assigns every pixel in an image to a speciﬁc category. For example, deep learning can 101

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be used to annotate the locations of roads in aerial photographs (Mnih and Hinton, 2010). The output need not have its form mirror the structure of the input as closely as in these annotation-style tasks. For example, in image captioning, the computer program observes an image and outputs a natural language sentence describing the image (Kiros et al., 2014a,b; Mao et al., 2015; Vinyals et al., 2015b; Donahue et al., 2014; Karpathy and Li, 2015; Fang et al., 2015; Xu et al., 2015). These tasks are called structured output tasks because the program must output several values that are all tightly inter-related. For example, the words produced by an image captioning program must form a valid sentence. • Anomaly detection: In this type of task, the computer program sifts through a set of events or objects, and ﬂags some of them as being unusual or atypical. An example of an anomaly detection task is credit card fraud detection. By modeling your purchasing habits, a credit card company can detect misuse of your cards. If a thief steals your credit card or credit card information, the thief’s purchases will often come from a diﬀerent probability distribution over purchase types than your own. The credit card company can prevent fraud by placing a hold on an account as soon as that card has been used for an uncharacteristic purchase. See Chandola et al. (2009) for a survey of anomaly detection methods. • Synthesis and sampling: In this type of task, the machine learning algorithm is asked to generate new examples that are similar to those in the training data. Synthesis and sampling via machine learning can be useful for media applications where it can be expensive or boring for an artist to generate large volumes of content by hand. For example, video games can automatically generate textures for large objects or landscapes, rather than requiring an artist to manually label each pixel (Luo et al., 2013). In some cases, we want the sampling or synthesis procedure to generate some speciﬁc kind of output given the input. For example, in a speech synthesis task, we provide a written sentence and ask the program to emit an audio waveform containing a spoken version of that sentence. This is a kind of structured output task, but with the added qualiﬁcation that there is no single correct output for each input, and we explicitly desire a large amount of variation in the output, in order for the output to seem more natural and realistic. • Imputation of missing values: In this type of task, the machine learning algorithm is given a new example x ∈ R n, but with some entries xi of x missing. The algorithm must provide a prediction of the values of the missing entries. 102

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• Denoising: In this type of task, the machine learning algorithm is given in input a corrupted example x˜ ∈ Rn obtained by an unknown corruption process from a clean example x ∈ R n. The learner must predict the clean example x from its corrupted version x, ˜ or more generally predict the conditional ˜ probability distribution p(x | x). • Density estimation or probability mass function estimation: In the density estimation problem, the machine learning algorithm is asked to learn a function pmodel : Rn → R, where pmodel (x) can be interpreted as a probability density function (if x is continuous) or a probability mass function (if x is discrete) on the space that the examples were drawn from. To do such a task well (we will specify exactly what that means when we discuss performance measures P ), the algorithm needs to learn the structure of the data it has seen. It must know where examples cluster tightly and where they are unlikely to occur. Most of the tasks described above require that the learning algorithm has at least implicitly captured the structure of the probability distribution. Density estimation allows us to explicitly capture that distribution. In principle, we can then perform computations on that distribution in order to solve the other tasks as well. For example, if we have performed density estimation to obtain a probability distribution p(x), we can use that distribution to solve the missing value imputation task. If a value x i is missing and all of the other values, denoted x−i, are given, then we know the distribution over it is given by p(xi | x −i). In practice, density estimation does not always allow us to solve all of these related tasks, because in many cases the required operations on p(x) are computationally intractable. Of course, many other tasks and types of tasks are possible. The types of tasks we list here are intended only to provide examples of what machine learning can do, not to deﬁne a rigid taxonomy of tasks.

5.1.2

The Performance Measure, P

In order to evaluate the abilities of a machine learning algorithm, we must design a quantitative measure of its performance. Usually this performance measure P is speciﬁc to the task T being carried out by the system. For tasks such as classiﬁcation, classiﬁcation with missing inputs, and transcription, we often measure the accuracy of the model. Accuracy is just the proportion of examples for which the model produces the correct output. We can also obtain 103

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equivalent information by measuring the error rate, the proportion of examples for which the model produces an incorrect output. We often refer to the error rate as the expected 0-1 loss. The 0-1 loss on a particular example is 0 if it is correctly classiﬁed and 1 if it is not. For tasks such as density estimation, it does not make sense to measure accuracy, error rate, or any other kind of 0-1 loss. Instead, we must use a diﬀerent performance metric that gives the model a continuous-valued score for each example. The most common approach is to report the average log-probability the model assigns to some examples. Usually we are interested in how well the machine learning algorithm performs on data that it has not seen before, since this determines how well it will work when deployed in the real world. We therefore evaluate these performance measures using a test set of data that is separate from the data used for training the machine learning system. The choice of performance measure may seem straightforward and objective, but it is often diﬃcult to choose a performance measure that corresponds well to the desired behavior of the system. In some cases, this is because it is diﬃcult to decide what should be measured. For example, when performing a transcription task, should we measure the accuracy of the system at transcribing entire sequences, or should we use a more ﬁne-grained performance measure that gives partial credit for getting some elements of the sequence correct? When performing a regression task, should we penalize the system more if it frequently makes medium-sized mistakes or if it rarely makes very large mistakes? These kinds of design choices depend on the application. In other cases, we know what quantity we would ideally like to measure, but measuring it is impractical. For example, this arises frequently in the context of density estimation. Many of the best probabilistic models represent probability distributions only implicitly. Computing the actual probability value assigned to a speciﬁc point in space in many such models is intractable. In these cases, one must design an alternative criterion that still corresponds to the design objectives, or design a good approximation to the desired criterion.

5.1.3

The Experience, E

Machine learning algorithms can be broadly categorized as unsupervised or supervised by what kind of experience they are allowed to have during the learning process. Most of the learning algorithms in this book can be understood as being allowed to experience an entire dataset. A dataset is a collection of many examples, as 104

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deﬁned in Sec. 5.1.1. Sometimes we will also call examples data points. One of the oldest datasets studied by statisticians and machine learning researchers is the Iris dataset (Fisher, 1936). It is a collection of measurements of diﬀerent parts of 150 iris plants. Each individual plant corresponds to one example. The features within each example are the measurements of each of the parts of the plant: the sepal length, sepal width, petal length and petal width. The dataset also records which species each plant belonged to. Three diﬀerent species are represented in the dataset. Unsupervised learning algorithms experience a dataset containing many features, then learn useful properties of the structure of this dataset. In the context of deep learning, we usually want to learn the entire probability distribution that generated a dataset, whether explicitly as in density estimation or implicitly for tasks like synthesis or denoising. Some other unsupervised learning algorithms perform other roles, like clustering, which consists of dividing the dataset into clusters of similar examples. Supervised learning algorithms experience a dataset containing features, but each example is also associated with a label or target. For example, the Iris dataset is annotated with the species of each iris plant. A supervised learning algorithm can study the Iris dataset and learn to classify iris plants into three diﬀerent species based on their measurements. Roughly speaking, unsupervised learning involves observing several examples of a random vector x, and attempting to implicitly or explicitly learn the probability distribution p(x), or some interesting properties of that distribution, while supervised learning involves observing several examples of a random vector x and an associated value or vector y, and learning to predict y from x, usually by estimating p(y | x ). The term supervised learning originates from the view of the target y being provided by an instructor or teacher who shows the machine learning system what to do. In unsupervised learning, there is no instructor or teacher, and the algorithm must learn to make sense of the data without this guide. Unsupervised learning and supervised learning are not formally deﬁned terms. The lines between them are often blurred. Many machine learning technologies can be used to perform both tasks. For example, the chain rule of probability states that for a vector x ∈ Rn, the joint distribution can be decomposed as p(x) =

n i=1

p(xi | x 1, . . . , xi−1 ).

(5.1)

This decomposition means that we can solve the ostensibly unsupervised problem of modeling p(x) by splitting it into n supervised learning problems. Alternatively, we 105

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can solve the supervised learning problem of learning p(y | x) by using traditional unsupervised learning technologies to learn the joint distribution p(x, y) and inferring p(x, y) p(y | x) = (5.2) ) . p (x , y y

Though unsupervised learning and supervised learning are not completely formal or distinct concepts, they do help to roughly categorize some of the things we do with machine learning algorithms. Traditionally, people refer to regression, classiﬁcation and structured output problems as supervised learning. Density estimation in support of other tasks is usually considered unsupervised learning.

Other variants of the learning paradigm are possible. For example, in semisupervised learning, some examples include a supervision target but others do not. In multi-instance learning, an entire collection of examples is labeled as containing or not containing an example of a class, but the individual members of the collection are not labeled. For a recent example of multi-instance learning with deep models, see Kotzias et al. (2015). Some machine learning algorithms do not just experience a ﬁxed dataset. For example, reinforcement learning algorithms interact with an environment, so there is a feedback loop between the learning system and its experiences. Such algorithms are beyond the scope of this book. Please see Sutton and Barto (1998) or Bertsekas and Tsitsiklis (1996) for information about reinforcement learning, and Mnih et al. (2013) for the deep learning approach to reinforcement learning. Most machine learning algorithms simply experience a dataset. A dataset can be described in many ways. In all cases, a dataset is a collection of examples, which are in turn collections of features. One common way of describing a dataset is with a design matrix. A design matrix is a matrix containing a diﬀerent example in each row. Each column of the matrix corresponds to a diﬀerent feature. For instance, the Iris dataset contains 150 examples with four features for each example. This means we can represent the dataset with a design matrix X ∈ R150×4 , where Xi,1 is the sepal length of plant i, X i,2 is the sepal width of plant i, etc. We will describe most of the learning algorithms in this book in terms of how they operate on design matrix datasets. Of course, to describe a dataset as a design matrix, it must be possible to describe each example as a vector, and each of these vectors must be the same size. This is not always possible. For example, if you have a collection of photographs with diﬀerent widths and heights, then diﬀerent photographs will contain diﬀerent numbers of pixels, so not all of the photographs may be described with the same length of vector. Sec. 9.7 and Chapter 10 describe how to handle diﬀerent types 106

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of such heterogeneous data. In cases like these, rather than describing the dataset as a matrix with m rows, we will describe it as a set containing m elements: {x(1) , x(2) , . . . , x (m)}. This notation does not imply that any two example vectors x(i) and x(j) have the same size. In the case of supervised learning, the example contains a label or target as well as a collection of features. For example, if we want to use a learning algorithm to perform object recognition from photographs, we need to specify which object appears in each of the photos. We might do this with a numeric code, with 0 signifying a person, 1 signifying a car, 2 signifying a cat, etc. Often when working with a dataset containing a design matrix of feature observations X, we also provide a vector of labels y , with yi providing the label for example i. Of course, sometimes the label may be more than just a single number. For example, if we want to train a speech recognition system to transcribe entire sentences, then the label for each example sentence is a sequence of words. Just as there is no formal deﬁnition of supervised and unsupervised learning, there is no rigid taxonomy of datasets or experiences. The structures described here cover most cases, but it is always possible to design new ones for new applications.

5.1.4

Example: Linear Regression

Our deﬁnition of a machine learning algorithm as an algorithm that is capable of improving a computer program’s performance at some task via experience is somewhat abstract. To make this more concrete, we present an example of a simple machine learning algorithm: linear regression. We will return to this example repeatedly as we introduce more machine learning concepts that help to understand its behavior. As the name implies, linear regression solves a regression problem. In other words, the goal is to build a system that can take a vector x ∈ Rn as input and predict the value of a scalar y ∈ R as its output. In the case of linear regression, the output is a linear function of the input. Let yˆ be the value that our model predicts y should take on. We deﬁne the output to be yˆ = w x

(5.3)

where w ∈ R n is a vector of parameters. Parameters are values that control the behavior of the system. In this case, wi is the coeﬃcient that we multiply by feature x i before summing up the contributions from all the features. We can think of w as a set of weights that determine how each feature aﬀects the prediction. If a feature xi receives a positive weight wi , 107

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then increasing the value of that feature increases the value of our prediction yˆ. If a feature receives a negative weight, then increasing the value of that feature decreases the value of our prediction. If a feature’s weight is large in magnitude, then it has a large eﬀect on the prediction. If a feature’s weight is zero, it has no eﬀect on the prediction. We thus have a deﬁnition of our task T : to predict y from x by outputting yˆ = w x. Next we need a deﬁnition of our performance measure, P . Suppose that we have a design matrix of m example inputs that we will not use for training, only for evaluating how well the model performs. We also have a vector of regression targets providing the correct value of y for each of these examples. Because this dataset will only be used for evaluation, we call it the test set. We refer to the design matrix of inputs as X(test) and the vector of regression targets as y (test) . One way of measuring the performance of the model is to compute the mean squared error of the model on the test set. If yˆ(test) gives the predictions of the model on the test set, then the mean squared error is given by MSEtest =

1 (test) − y(test)) 2i . (yˆ m i

(5.4)

Intuitively, one can see that this error measure decreases to 0 when yˆ (test) = y(test). We can also see that MSEtest =

1 (test) ||ˆy − y (test) ||22 , m

(5.5)

so the error increases whenever the Euclidean distance between the predictions and the targets increases. To make a machine learning algorithm, we need to design an algorithm that will improve the weights w in a way that reduces MSEtest when the algorithm is allowed to gain experience by observing a training set (X(train), y (train) ). One intuitive way of doing this (which we will justify later, in Sec. 5.5.1) is just to minimize the mean squared error on the training set, MSEtrain . To minimize MSE train, we can simply solve for where its gradient is 0: ∇w MSEtrain = 0 1 ⇒ ∇ w ||yˆ (train) − y (train)|| 22 = 0 m 1 ⇒ ∇ w ||X(train)w − y (train)|| 22 = 0 m 108

(5.6) (5.7) (5.8)

CHAPTER 5. MACHINE LEARNING BASICS

3

Linear regression example

0.55 0.50

2 MSE (train)

y

1 0 −1 −2 −3

Optimization of w

0.45 0.40 0.35 0.30 0.25

−1.0 −0.5

0.0 x1

0.5

0.20

1.0

0.5

1.0 w1

1.5

Figure 5.1: A linear regression problem, with a training set consisting of ten data points, each containing one feature. Because there is only one feature, the weight vector w contains only a single parameter to learn, w 1 . (Left) Observe that linear regression learns to set w 1 such that the line y = w 1 x comes as close as possible to passing through all the training points. (Right) The plotted point indicates the value of w 1 found by the normal equations, which we can see minimizes the mean squared error on the training set.

(train) (train) (train) (train) X =0 ⇒ ∇w X w−y w−y

(5.9)

(train) (train) (train) (train) (train) (train) X y +y y =0 ⇒ ∇w w X w − 2w X (5.10) (train) (train) (train) (train) X y =0 (5.11) ⇒ 2X w − 2X −1 X(train) y(train) (5.12) ⇒ w = X (train) X(train)

The system of equations whose solution is given by Eq. 5.12 is known as the normal equations. Evaluating Eq. 5.12 constitutes a simple learning algorithm. For an example of the linear regression learning algorithm in action, see Fig. 5.1. It is worth noting that the term linear regression is often used to refer to a slightly more sophisticated model with one additional parameter—an intercept term b. In this model yˆ = w x + b (5.13) so the mapping from parameters to predictions is still a linear function but the mapping from features to predictions is now an aﬃne function. This extension to aﬃne functions means that the plot of the model’s predictions still looks like a line, but it need not pass through the origin. Instead of adding the bias parameter b, one can continue to use the model with only weights but augment x with an 109

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extra entry that is always set to 1. The weight corresponding to the extra 1 entry plays the role of the bias parameter. We will frequently use the term “linear” when referring to aﬃne functions throughout this book. The intercept term b is often called the bias parameter of the aﬃne transformation. This terminology derives from the point of view that the output of the transformation is biased toward being b in the absence of any input. This term is diﬀerent from the idea of a statistical bias, in which a statistical estimation algorithm’s expected estimate of a quantity is not equal to the true quantity. Linear regression is of course an extremely simple and limited learning algorithm, but it provides an example of how a learning algorithm can work. In the subsequent sections we will describe some of the basic principles underlying learning algorithm design and demonstrate how these principles can be used to build more complicated learning algorithms.

5.2

Capacity, Overﬁtting and Underﬁtting

The central challenge in machine learning is that we must perform well on new, previously unseen inputs—not just those on which our model was trained. The ability to perform well on previously unobserved inputs is called generalization. Typically, when training a machine learning model, we have access to a training set, we can compute some error measure on the training set called the training error, and we reduce this training error. So far, what we have described is simply an optimization problem. What separates machine learning from optimization is that we want the generalization error, also called the test error, to be low as well. The generalization error is deﬁned as the expected value of the error on a new input. Here the expectation is taken across diﬀerent possible inputs, drawn from the distribution of inputs we expect the system to encounter in practice. We typically estimate the generalization error of a machine learning model by measuring its performance on a test set of examples that were collected separately from the training set. In our linear regression example, we trained the model by minimizing the training error, 1 ||X (train) w − y(train) || 22, (5.14) ( train ) m but we actually care about the test error,

1 ||X (test) w m(test)

− y(test) ||22.

How can we aﬀect performance on the test set when we get to observe only the training set? The ﬁeld of statistical learning theory provides some answers. If the 110

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training and the test set are collected arbitrarily, there is indeed little we can do. If we are allowed to make some assumptions about how the training and test set are collected, then we can make some progress. The train and test data are generated by a probability distribution over datasets called the data generating process. We typically make a set of assumptions known collectively as the i.i.d. assumptions These assumptions are that the examples in each dataset are independent from each other, and that the train set and test set are identically distributed, drawn from the same probability distribution as each other. This assumption allows us to describe the data generating process with a probability distribution over a single example. The same distribution is then used to generate every train example and every test example. We call that shared underlying distribution the data generating distribution, denoted p data. This probabilistic framework and the i.i.d. assumptions allow us to mathematically study the relationship between training error and test error. One immediate connection we can observe between the training and test error is that the expected training error of a randomly selected model is equal to the expected test error of that model. Suppose we have a probability distribution p(x, y) and we sample from it repeatedly to generate the train set and the test set. For some ﬁxed value w, the expected training set error is exactly the same as the expected test set error, because both expectations are formed using the same dataset sampling process. The only diﬀerence between the two conditions is the name we assign to the dataset we sample. Of course, when we use a machine learning algorithm, we do not ﬁx the parameters ahead of time, then sample both datasets. We sample the training set, then use it to choose the parameters to reduce training set error, then sample the test set. Under this process, the expected test error is greater than or equal to the expected value of training error. The factors determining how well a machine learning algorithm will perform are its ability to: 1. Make the training error small. 2. Make the gap between training and test error small. These two factors correspond to the two central challenges in machine learning: underﬁtting and overﬁtting. Underﬁtting occurs when the model is not able to obtain a suﬃciently low error value on the training set. Overﬁtting occurs when the gap between the training error and test error is too large. We can control whether a model is more likely to overﬁt or underﬁt by altering its capacity. Informally, a model’s capacity is its ability to ﬁt a wide variety of 111

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functions. Models with low capacity may struggle to ﬁt the training set. Models with high capacity can overﬁt by memorizing properties of the training set that do not serve them well on the test set. One way to control the capacity of a learning algorithm is by choosing its hypothesis space, the set of functions that the learning algorithm is allowed to select as being the solution. For example, the linear regression algorithm has the set of all linear functions of its input as its hypothesis space. We can generalize linear regression to include polynomials, rather than just linear functions, in its hypothesis space. Doing so increases the model’s capacity. A polynomial of degree one gives us the linear regression model with which we are already familiar, with prediction yˆ = b + wx.

(5.15)

By introducing x2 as another feature provided to the linear regression model, we can learn a model that is quadratic as a function of x: yˆ = b + w 1x + w2 x2 .

(5.16)

Though this model implements a quadratic function of its input, the output is still a linear function of the parameters, so we can still use the normal equations to train the model in closed form. We can continue to add more powers of x as additional features, for example to obtain a polynomial of degree 9: yˆ = b +

9

wi xi.

(5.17)

i=1

Machine learning algorithms will generally perform best when their capacity is appropriate in regard to the true complexity of the task they need to perform and the amount of training data they are provided with. Models with insuﬃcient capacity are unable to solve complex tasks. Models with high capacity can solve complex tasks, but when their capacity is higher than needed to solve the present task they may overﬁt. Fig. 5.2 shows this principle in action. We compare a linear, quadratic and degree-9 predictor attempting to ﬁt a problem where the true underlying function is quadratic. The linear function is unable to capture the curvature in the true underlying problem, so it underﬁts. The degree-9 predictor is capable of representing the correct function, but it is also capable of representing inﬁnitely many other functions that pass exactly through the training points, because we have more 112

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parameters than training examples. We have little chance of choosing a solution that generalizes well when so many wildly diﬀerent solutions exist. In this example, the quadratic model is perfectly matched to the true structure of the task so it generalizes well to new data.

Figure 5.2: We ﬁt three models to this example training set. The training data was generated synthetically, by randomly sampling x values and choosing y deterministically by evaluating a quadratic function. (Left) A linear function ﬁt to the data suﬀers from underﬁtting—it cannot capture the curvature that is present in the data. (Center) A quadratic function ﬁt to the data generalizes well to unseen points. It does not suﬀer from a signiﬁcant amount of overﬁtting or underﬁtting. (Right) A polynomial of degree 9 ﬁt to the data suﬀers from overﬁtting. Here we used the Moore-Penrose pseudoinverse to solve the underdetermined normal equations. The solution passes through all of the training points exactly, but we have not been lucky enough for it to extract the correct structure. It now has a deep valley in between two training points that does not appear in the true underlying function. It also increases sharply on the left side of the data, while the true function decreases in this area.

So far we have only described changing a model’s capacity by changing the number of input features it has (and simultaneously adding new parameters associated with those features). There are in fact many ways of changing a model’s capacity. Capacity is not determined only by the choice of model. The model speciﬁes which family of functions the learning algorithm can choose from when varying the parameters in order to reduce a training objective. This is called the representational capacity of the model. In many cases, ﬁnding the best function within this family is a very diﬃcult optimization problem. In practice, the learning algorithm does not actually ﬁnd the best function, but merely one that signiﬁcantly reduces the training error. These additional limitations, such as the imperfection 113

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of the optimization algorithm, mean that the learning algorithm’s eﬀective capacity may be less than the representational capacity of the model family. Our modern ideas about improving the generalization of machine learning models are reﬁnements of thought dating back to philosophers at least as early as Ptolemy. Many early scholars invoke a principle of parsimony that is now most widely known as Occam’s razor (c. 1287-1347). This principle states that among competing hypotheses that explain known observations equally well, one should choose the “simplest” one. This idea was formalized and made more precise in the 20th century by the founders of statistical learning theory (Vapnik and Chervonenkis, 1971; Vapnik, 1982; Blumer et al., 1989; Vapnik, 1995). Statistical learning theory provides various means of quantifying model capacity. Among these, the most well-known is the Vapnik-Chervonenkis dimension, or VC dimension. The VC dimension measures the capacity of a binary classiﬁer. The VC dimension is deﬁned as being the largest possible value of m for which there exists a training set of m diﬀerent x points that the classiﬁer can label arbitrarily. Quantifying the capacity of the model allows statistical learning theory to make quantitative predictions. The most important results in statistical learning theory show that the discrepancy between training error and generalization error is bounded from above by a quantity that grows as the model capacity grows but shrinks as the number of training examples increases (Vapnik and Chervonenkis, 1971; Vapnik, 1982; Blumer et al., 1989; Vapnik, 1995). These bounds provide intellectual justiﬁcation that machine learning algorithms can work, but they are rarely used in practice when working with deep learning algorithms. This is in part because the bounds are often quite loose and in part because it can be quite diﬃcult to determine the capacity of deep learning algorithms. The problem of determining the capacity of a deep learning model is especially diﬃcult because the eﬀective capacity is limited by the capabilities of the optimization algorithm, and we have little theoretical understanding of the very general non-convex optimization problems involved in deep learning. We must remember that while simpler functions are more likely to generalize (to have a small gap between training and test error) we must still choose a suﬃciently complex hypothesis to achieve low training error. Typically, training error decreases until it asymptotes to the minimum possible error value as model capacity increases (assuming the error measure has a minimum value). Typically, generalization error has a U-shaped curve as a function of model capacity. This is illustrated in Fig. 5.3. To reach the most extreme case of arbitrarily high capacity, we introduce the concept of non-parametric models. So far, we have seen only parametric 114

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Training error Generalization error

Error

Underﬁtting zone Overﬁtting zone

Generalization gap 0

Optimal Capacity Capacity

Figure 5.3: Typical relationship between capacity and error. Training and test error behave diﬀerently. At the left end of the graph, training error and generalization error are both high. This is the underﬁtting regime. As we increase capacity, training error decreases, but the gap between training and generalization error increases. Eventually, the size of this gap outweighs the decrease in training error, and we enter the overﬁtting regime, where capacity is too large, above the optimal capacity.

models, such as linear regression. Parametric models learn a function described by a parameter vector whose size is ﬁnite and ﬁxed before any data is observed. Non-parametric models have no such limitation. Sometimes, non-parametric models are just theoretical abstractions (such as an algorithm that searches over all possible probability distributions) that cannot be implemented in practice. However, we can also design practical non-parametric models by making their complexity a function of the training set size. One example of such an algorithm is nearest neighbor regression. Unlike linear regression, which has a ﬁxed-length vector of weights, the nearest neighbor regression model simply stores the X and y from the training set. When asked to classify a test point x, the model looks up the nearest entry in the training set and returns the associated regression target. In other words, yˆ = yi where i = arg min ||Xi,: − x||22. The algorithm can also be generalized to distance metrics other than the L2 norm, such as learned distance metrics (Goldberger et al., 2005). If the algorithm is allowed to break ties by averaging the yi values for all X i,: that are tied for nearest, then this algorithm is able to achieve the minimum possible training error (which might be greater than zero, if two identical inputs are associated with diﬀerent outputs) on any regression dataset. Finally, we can also create a non-parametric learning algorithm by wrapping a parametric learning algorithm inside another algorithm that increases the number 115

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of parameters as needed. For example, we could imagine an outer loop of learning that changes the degree of the polynomial learned by linear regression on top of a polynomial expansion of the input. The ideal model is an oracle that simply knows the true probability distribution that generates the data. Even such a model will still incur some error on many problems, because there may still be some noise in the distribution. In the case of supervised learning, the mapping from x to y may be inherently stochastic, or y may be a deterministic function that involves other variables besides those included in x. The error incurred by an oracle making predictions from the true distribution p(x, y) is called the Bayes error. Training and generalization error vary as the size of the training set varies. Expected generalization error can never increase as the number of training examples increases. For non-parametric models, more data yields better generalization until the best possible error is achieved. Any ﬁxed parametric model with less than optimal capacity will asymptote to an error value that exceeds the Bayes error. See Fig. 5.4 for an illustration. Note that it is possible for the model to have optimal capacity and yet still have a large gap between training and generalization error. In this situation, we may be able to reduce this gap by gathering more training examples.

5.2.1

The No Free Lunch Theorem

Learning theory claims that a machine learning algorithm can generalize well from a ﬁnite training set of examples. This seems to contradict some basic principles of logic. Inductive reasoning, or inferring general rules from a limited set of examples, is not logically valid. To logically infer a rule describing every member of a set, one must have information about every member of that set. In part, machine learning avoids this problem by oﬀering only probabilistic rules, rather than the entirely certain rules used in purely logical reasoning. Machine learning promises to ﬁnd rules that are probably correct about most members of the set they concern. Unfortunately, even this does not resolve the entire problem. The no free lunch theorem for machine learning (Wolpert, 1996) states that, averaged over all possible data generating distributions, every classiﬁcation algorithm has the same error rate when classifying previously unobserved points. In other words, in some sense, no machine learning algorithm is universally any better than any other. The most sophisticated algorithm we can conceive of has the same average performance (over all possible tasks) as merely predicting that every point belongs to the same class. 116

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Figure 5.4: The eﬀect of the training dataset size on the train and test error, as well as on the optimal model capacity. We constructed a synthetic regression problem based on adding moderate amount of noise to a degree 5 polynomial, generated a single test set, and then generated several diﬀerent sizes of training set. For each size, we generated 40 diﬀerent training sets in order to plot error bars showing 95% conﬁdence intervals. (Top) The MSE on the train and test set for two diﬀerent models: a quadratic model, and a model with degree chosen to minimize the test error. Both are ﬁt in closed form. For the quadratic model, the training error increases as the size of the training set increases. This is because larger datasets are harder to ﬁt. Simultaneously, the test error decreases, because fewer incorrect hypotheses are consistent with the training data. The quadratic model does not have enough capacity to solve the task, so its test error asymptotes to a high value. The test error at optimal capacity asymptotes to the Bayes error. The training error can fall below the Bayes error, due to the ability of the training algorithm to memorize speciﬁc instances of the training set. As the training size increases to inﬁnity, the training error of any ﬁxed-capacity model (here, the quadratic model) must rise to at least the Bayes error. (Bottom) As the training set size increases, the optimal capacity (shown here as the degree of the optimal polynomial regressor) increases. The optimal capacity plateaus after reaching suﬃcient complexity to solve the task. 117

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Fortunately, these results hold only when we average over all possible data generating distributions. If we make assumptions about the kinds of probability distributions we encounter in real-world applications, then we can design learning algorithms that perform well on these distributions. This means that the goal of machine learning research is not to seek a universal learning algorithm or the absolute best learning algorithm. Instead, our goal is to understand what kinds of distributions are relevant to the “real world” that an AI agent experiences, and what kinds of machine learning algorithms perform well on data drawn from the kinds of data generating distributions we care about.

5.2.2

Regularization

The no free lunch theorem implies that we must design our machine learning algorithms to perform well on a speciﬁc task. We do so by building a set of preferences into the learning algorithm. When these preferences are aligned with the learning problems we ask the algorithm to solve, it performs better. So far, the only method of modifying a learning algorithm we have discussed is to increase or decrease the model’s capacity by adding or removing functions from the hypothesis space of solutions the learning algorithm is able to choose. We gave the speciﬁc example of increasing or decreasing the degree of a polynomial for a regression problem. The view we have described so far is oversimpliﬁed. The behavior of our algorithm is strongly aﬀected not just by how large we make the set of functions allowed in its hypothesis space, but by the speciﬁc identity of those functions. The learning algorithm we have studied so far, linear regression, has a hypothesis space consisting of the set of linear functions of its input. These linear functions can be very useful for problems where the relationship between inputs and outputs truly is close to linear. They are less useful for problems that behave in a very nonlinear fashion. For example, linear regression would not perform very well if we tried to use it to predict sin(x) from x. We can thus control the performance of our algorithms by choosing what kind of functions we allow them to draw solutions from, as well as by controlling the amount of these functions. We can also give a learning algorithm a preference for one solution in its hypothesis space to another. This means that both functions are eligible, but one is preferred. The unpreferred solution be chosen only if it ﬁts the training data signiﬁcantly better than the preferred solution. For example, we can modify the training criterion for linear regression to include weight decay. To perform linear regression with weight decay, we minimize 118

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a sum comprising both the mean squared error on the training and a criterion J(w) that expresses a preference for the weights to have smaller squared L2 norm. Speciﬁcally, J(w) = MSEtrain + λw w, (5.18) where λ is a value chosen ahead of time that controls the strength of our preference for smaller weights. When λ = 0, we impose no preference, and larger λ forces the weights to become smaller. Minimizing J (w ) results in a choice of weights that make a tradeoﬀ between ﬁtting the training data and being small. This gives us solutions that have a smaller slope, or put weight on fewer of the features. As an example of how we can control a model’s tendency to overﬁt or underﬁt via weight decay, we can train a high-degree polynomial regression model with diﬀerent values of λ. See Fig. 5.5 for the results.

Figure 5.5: We ﬁt a high-degree polynomial regression model to our example training set from Fig. 5.2. The true function is quadratic, but here we use only models with degree 9. We vary the amount of weight decay to prevent these high-degree models from overﬁtting. (Left) With very large λ, we can force the model to learn a function with no slope at all. This underﬁts because it can only represent a constant function. (Center) With a medium value of λ, the learning algorithm recovers a curve with the right general shape. Even though the model is capable of representing functions with much more complicated shape, weight decay has encouraged it to use a simpler function described by smaller coeﬃcients. (Right) With weight decay approaching zero (i.e., using the Moore-Penrose pseudoinverse to solve the underdetermined problem with minimal regularization), the degree-9 polynomial overﬁts signiﬁcantly, as we saw in Fig. 5.2.

More generally, we can regularize a model that learns a function f(x; θ) by adding a penalty called a regularizer to the cost function. In the case of weight decay, the regularizer is Ω(w) = w w. In Chapter 7, we will see that many other 119

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regularizers are possible. Expressing preferences for one function over another is a more general way of controlling a model’s capacity than including or excluding members from the hypothesis space. We can think of excluding a function from a hypothesis space as expressing an inﬁnitely strong preference against that function. In our weight decay example, we expressed our preference for linear functions deﬁned with smaller weights explicitly, via an extra term in the criterion we minimize. There are many other ways of expressing preferences for diﬀerent solutions, both implicitly and explicitly. Together, these diﬀerent approaches are known as regularization. Regularization is any modiﬁcation we make to a learning algorithm that is intended to reduce its generalization error but not its training error. Regularization is one of the central concerns of the ﬁeld of machine learning, rivaled in its importance only by optimization. The no free lunch theorem has made it clear that there is no best machine learning algorithm, and, in particular, no best form of regularization. Instead we must choose a form of regularization that is well-suited to the particular task we want to solve. The philosophy of deep learning in general and this book in particular is that a very wide range of tasks (such as all of the intellectual tasks that people can do) may all be solved eﬀectively using very general-purpose forms of regularization.

5.3

Hyperparameters and Validation Sets

Most machine learning algorithms have several settings that we can use to control the behavior of the learning algorithm. These settings are called hyperparameters. The values of hyperparameters are not adapted by the learning algorithm itself (though we can design a nested learning procedure where one learning algorithm learns the best hyperparameters for another learning algorithm). In the polynomial regression example we saw in Fig. 5.2, there is a single hyperparameter: the degree of the polynomial, which acts as a capacity hyperparameter. The λ value used to control the strength of weight decay is another example of a hyperparameter. Sometimes a setting is chosen to be a hyperparameter that the learning algorithm does not learn because it is diﬃcult to optimize. More frequently, we do not learn the hyperparameter because it is not appropriate to learn that hyperparameter on the training set. This applies to all hyperparameters that control model capacity. If learned on the training set, such hyperparameters would always 120

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choose the maximum possible model capacity, resulting in overﬁtting (refer to Fig. 5.3). For example, we can always ﬁt the training set better with a higher degree polynomial and a weight decay setting of λ = 0 than we could with a lower degree polynomial and a positive weight decay setting. To solve this problem, we need a validation set of examples that the training algorithm does not observe. Earlier we discussed how a held-out test set, composed of examples coming from the same distribution as the training set, can be used to estimate the generalization error of a learner, after the learning process has completed. It is important that the test examples are not used in any way to make choices about the model, including its hyperparameters. For this reason, no example from the test set can be used in the validation set. Therefore, we always construct the validation set from the training data. Speciﬁcally, we split the training data into two disjoint subsets. One of these subsets is used to learn the parameters. The other subset is our validation set, used to estimate the generalization error during or after training, allowing for the hyperparameters to be updated accordingly. The subset of data used to learn the parameters is still typically called the training set, even though this may be confused with the larger pool of data used for the entire training process. The subset of data used to guide the selection of hyperparameters is called the validation set. Typically, one uses about 80% of the training data for training and 20% for validation. Since the validation set is used to “train” the hyperparameters, the validation set error will underestimate the generalization error, though typically by a smaller amount than the training error. After all hyperparameter optimization is complete, the generalization error may be estimated using the test set. In practice, when the same test set has been used repeatedly to evaluate performance of diﬀerent algorithms over many years, and especially if we consider all the attempts from the scientiﬁc community at beating the reported state-ofthe-art performance on that test set, we end up having optimistic evaluations with the test set as well. Benchmarks can thus become stale and then do not reﬂect the true ﬁeld performance of a trained system. Thankfully, the community tends to move on to new (and usually more ambitious and larger) benchmark datasets.

5.3.1

Cross-Validation

Dividing the dataset into a ﬁxed training set and a ﬁxed test set can be problematic if it results in the test set being small. A small test set implies statistical uncertainty around the estimated average test error, making it diﬃcult to claim that algorithm A works better than algorithm B on the given task. 121

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When the dataset has hundreds of thousands of examples or more, this is not a serious issue. When the dataset is too small, there are alternative procedures, which allow one to use all of the examples in the estimation of the mean test error, at the price of increased computational cost. These procedures are based on the idea of repeating the training and testing computation on diﬀerent randomly chosen subsets or splits of the original dataset. The most common of these is the k-fold cross-validation procedure, shown in Algorithm 5.1, in which a partition of the dataset is formed by splitting it into k non-overlapping subsets. The test error may then be estimated by taking the average test error across k trials. On trial i, the i-th subset of the data is used as the test set and the rest of the data is used as the training set. One problem is that there exist no unbiased estimators of the variance of such average error estimators (Bengio and Grandvalet, 2004), but approximations are typically used.

5.4

Estimators, Bias and Variance

The ﬁeld of statistics gives us many tools that can be used to achieve the machine learning goal of solving a task not only on the training set but also to generalize. Foundational concepts such as parameter estimation, bias and variance are useful to formally characterize notions of generalization, underﬁtting and overﬁtting.

5.4.1

Point Estimation

Point estimation is the attempt to provide the single “best” prediction of some quantity of interest. In general the quantity of interest can be a single parameter or a vector of parameters in some parametric model, such as the weights in our linear regression example in Sec. 5.1.4, but it can also be a whole function. In order to distinguish estimates of parameters from their true value, our ˆ convention will be to denote a point estimate of a parameter θ by θ. Let {x(1) , . . . , x(m) } be a set of m independent and identically distributed (i.i.d.) data points. A point estimator or statistic is any function of the data: θˆ m = g(x(1) , . . . , x(m) ).

(5.19)

The deﬁnition does not require that g return a value that is close to the true θ or even that the range of g is the same as the set of allowable values of θ. This deﬁnition of a point estimator is very general and allows the designer of an estimator great ﬂexibility. While almost any function thus qualiﬁes as an estimator, 122

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Algorithm 5.1 The k-fold cross-validation algorithm. It can be used to estimate generalization error of a learning algorithm A when the given dataset D is too small for a simple train/test or train/valid split to yield accurate estimation of generalization error, because the mean of a loss L on a small test set may have too high variance. The dataset D contains as elements the abstract examples z(i) (for the i-th example), which could stand for an (input,target) pair z(i) = (x(i) , y(i)) in the case of supervised learning, or for just an input z (i) = x(i) in the case of unsupervised learning. The algorithm returns the vector of errors e for each example in D, whose mean is the estimated generalization error. The errors on individual examples can be used to compute a conﬁdence interval around the mean (Eq. 5.47). While these conﬁdence intervals are not well-justiﬁed after the use of cross-validation, it is still common practice to use them to declare that algorithm A is better than algorithm B only if the conﬁdence interval of the error of algorithm A lies below and does not intersect the conﬁdence interval of algorithm B. Deﬁne KFoldXV(D, A, L, k): Require: D, the given dataset, with elements z(i) Require: A, the learning algorithm, seen as a function that takes a dataset as input and outputs a learned function Require: L, the loss function, seen as a function from a learned function f and an example z(i) ∈ D to a scalar ∈ R Require: k, the number of folds Split D into k mutually exclusive subsets Di , whose union is D. for i from 1 to k do fi = A(D\D i) for z (j) in D i do ej = L(fi, z (j)) end for end for Return e

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a good estimator is a function whose output is close to the true underlying θ that generated the training data. For now, we take the frequentist perspective on statistics. That is, we assume that the true parameter value θ is ﬁxed but unknown, while the point estimate θˆ is a function of the data. Since the data is drawn from a random process, any function of the data is random. Therefore θˆ is a random variable. Point estimation can also refer to the estimation of the relationship between input and target variables. We refer to these types of point estimates as function estimators. Function Estimation As we mentioned above, sometimes we are interested in performing function estimation (or function approximation). Here we are trying to predict a variable y given an input vector x . We assume that there is a function f(x) that describes the approximate relationship between y and x. For example, we may assume that y = f(x) + , where stands for the part of y that is not predictable from x. In function estimation, we are interested in approximating ˆ Function estimation is really just the same as f with a model or estimate f. estimating a parameter θ; the function estimator fˆ is simply a point estimator in function space. The linear regression example (discussed above in Sec. 5.1.4) and the polynomial regression example (discussed in Sec. 5.2) are both examples of scenarios that may be interpreted either as estimating a parameter w or estimating a function fˆ mapping from x to y. We now review the most commonly studied properties of point estimators and discuss what they tell us about these estimators.

5.4.2

Bias

The bias of an estimator is deﬁned as: bias(θˆm ) = E( θˆm ) − θ

(5.20)

where the expectation is over the data (seen as samples from a random variable) and θ is the true underlying value of θ used to deﬁne the data generating distribution. An estimator θˆm is said to be unbiased if bias(θˆm) = 0, which implies that E( θˆm ) = θ. An estimator θˆm is said to be asymptotically unbiased if lim m→∞ bias(θˆm ) = 0, which implies that lim m→∞ E( θˆm) = θ. Example: Bernoulli Distribution Consider a set of samples {x(1) , . . . , x(m) } that are independently and identically distributed according to a Bernoulli distri124

CHAPTER 5. MACHINE LEARNING BASICS

bution with mean θ: P (x(i); θ) = θ x

( i)

(1 − θ)(1−x

( i) )

.

(5.21)

A common estimator for the θ parameter of this distribution is the mean of the training samples: m 1 (i) ˆ θm = x . (5.22) m i=1

To determine whether this estimator is biased, we can substitute Eq. 5.22 into Eq. 5.20: bias(ˆθm ) = E[θˆm] − θ m 1 (i) −θ =E x m i=1 1 (i) E x = −θ m i=1

(5.23) (5.24)

m

m 1 1 (i) x (i) (1−x(i) ) = x θ (1 − θ) −θ m i=1 (i) x

=

1 m

m i=1

(5.25)

(5.26)

=0

(θ ) − θ

=θ−θ =0

(5.27) (5.28)

ˆ = 0, we say that our estimator θˆ is unbiased. Since bias( θ) Example: Gaussian Distribution Estimator of the Mean Now, consider a set of samples {x(1), . . . , x(m) } that are independently and identically distributed according to a Gaussian distribution p(x (i)) = N (x(i); µ, σ 2), where i ∈ {1, . . . , m}. Recall that the Gaussian probability density function is given by (i) − µ)2 1 1 (x p(x(i); µ, σ 2) = √ exp − . (5.29) 2 σ2 2πσ2 A common estimator of the Gaussian mean parameter is known as the sample mean: m 1 (i) µ ˆm = x (5.30) m i=1 125

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To determine the bias of the sample mean, we are again interested in calculating its expectation: bias(ˆ µ m ) = E[ˆ µm ] − µ m 1 (i) −µ =E x m i=1 m 1 (i) = −µ E x m i=1 m 1 = µ −µ m i=1 =µ−µ=0

(5.31) (5.32) (5.33) (5.34) (5.35)

Thus we ﬁnd that the sample mean is an unbiased estimator of Gaussian mean parameter. Example: Estimators of the Variance of a Gaussian Distribution As an example, we compare two diﬀerent estimators of the variance parameter σ 2 of a Gaussian distribution. We are interested in knowing if either estimator is biased. The ﬁrst estimator of σ 2 we consider is known as the sample variance: σ ˆ2m

m 2 1 (i) = x −µ ˆm , m i=1

(5.36)

ˆm is the sample mean, deﬁned above. More formally, we are interested in where µ computing 2 bias(ˆ σ 2m) = E[ˆ σm (5.37) ] − σ2 2 ]: We begin by evaluating the term E[ˆ σm m 2 1 2 E[ˆ σm ] =E x (i) − µ ˆm m i=1

=

m−1 2 σ m

(5.38) (5.39)

2 is −σ 2/m. Therefore, the ˆm Returning to Eq. 5.37, we conclude that the bias of σ sample variance is a biased estimator.

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The unbiased sample variance estimator 2 1 (i) = x −µ ˆm m−1 m

2 σ ˜m

(5.40)

i=1

provides an alternative approach. As the name suggests this estimator is unbiased. That is, we ﬁnd that E[˜ σ 2m ] = σ2: m 2 1 E[˜ σ2m] = E x (i) − µ (5.41) ˆm m−1 i=1 m 2 = E[ˆ σm ] (5.42) m−1 m m−1 2 = σ (5.43) m m−1 = σ 2.

(5.44)

We have two estimators: one is biased and the other is not. While unbiased estimators are clearly desirable, they are not always the “best” estimators. As we will see we often use biased estimators that possess other important properties.

5.4.3

Variance and Standard Error

Another property of the estimator that we might want to consider is how much we expect it to vary as a function of the data sample. Just as we computed the expectation of the estimator to determine its bias, we can compute its variance. The variance of an estimator is simply the variance ˆ Var( θ)

(5.45)

where the random variable is the training set. Alternately, the square root of the ˆ variance is called the standard error, denoted SE( θ). The variance or the standard error of an estimator provides a measure of how we would expect the estimate we compute from data to vary as we independently resample the dataset from the underlying data generating process. Just as we might like an estimator to exhibit low bias we would also like it to have relatively low variance. When we compute any statistic using a ﬁnite number of samples, our estimate of the true underlying parameter is uncertain, in the sense that we could have obtained other samples from the same distribution and their statistics would have 127

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been diﬀerent. The expected degree of variation in any estimator is a source of error that we want to quantify. The standard error of the mean is given by m 1 (i) σ SE(ˆ µ m) = Var[ x ]= √ , m i=1 m

(5.46)

where σ2 is the true variance of the samples x i. The standard error is often estimated by using an estimate of σ. Unfortunately, neither the square root of the sample variance nor the square root of the unbiased estimator of the variance provide an unbiased estimate of the standard deviation. Both approaches tend to underestimate the true standard deviation, but are still used in practice. The square root of the unbiased estimator of the variance is less of an underestimate. For large m, the approximation is quite reasonable. The standard error of the mean is very useful in machine learning experiments. We often estimate the generalization error by computing the sample mean of the error on the test set. The number of examples in the test set determines the accuracy of this estimate. Taking advantage of the central limit theorem, which tells us that the mean will be approximately distributed with a normal distribution, we can use the standard error to compute the probability that the true expectation falls in any chosen interval. For example, the 95% conﬁdence interval centered on the mean is µ ˆm is (ˆ µ m − 1.96SE(ˆ ˆm + 1.96SE(ˆ µm )), µ m ), µ

(5.47)

ˆ m and variance SE( µ ˆm ) 2. In machine under the normal distribution with mean µ learning experiments, it is common to say that algorithm A is better than algorithm B if the upper bound of the 95% conﬁdence interval for the error of algorithm A is less than the lower bound of the 95% conﬁdence interval for the error of algorithm B. Example: Bernoulli Distribution We once again consider a set of samples {x(1) , . . . , x (m)} drawn independently and identically from a Bernoulli distribution ( i) ( i) (recall P (x(i); θ) = θ x (1 − θ)(1−x ) ).This time we are interested in computing m 1 (i) the variance of the estimator ˆθm = m i=1 x .

Var θˆm = Var

128

m

1 (i) x m i=1

(5.48)

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m 1 = 2 Var x (i) m

=

1 m2

i=1 m i=1

θ(1 − θ)

(5.49) (5.50)

1 = 2 mθ(1 − θ) (5.51) m 1 (5.52) = θ(1 − θ) m The variance of the estimator decreases as a function of m, the number of examples in the dataset. This is a common property of popular estimators that we will return to when we discuss consistency (see Sec. 5.4.5).

5.4.4

Trading oﬀ Bias and Variance to Minimize Mean Squared Error

Bias and variance measure two diﬀerent sources of error in an estimator. Bias measures the expected deviation from the true value of the function or parameter. Variance on the other hand, provides a measure of the deviation from the expected estimator value that any particular sampling of the data is likely to cause. What happens when we are given a choice between two estimators, one with more bias and one with more variance? How do we choose between them? For example, imagine that we are interested in approximating the function shown in Fig. 5.2 and we are only oﬀered the choice between a model with large bias and one that suﬀers from large variance. How do we choose between them? The most common way to negotiate this trade-oﬀ is to use cross-validation. Empirically, cross-validation is highly successful on many real-world tasks. Alternatively, we can also compare the mean squared error (MSE) of the estimates: MSE = E[(ˆθm − θ) 2 ] = Bias( ˆθm) 2 + Var(θˆm )

(5.53) (5.54)

The MSE measures the overall expected deviation—in a squared error sense— between the estimator and the true value of the parameter θ. As is clear from Eq. 5.54, evaluating the MSE incorporates both the bias and the variance. Desirable estimators are those with small MSE and these are estimators that manage to keep both their bias and variance somewhat in check. The relationship between bias and variance is tightly linked to the machine learning concepts of capacity, underﬁtting and overﬁtting. In the case where gen129

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Underﬁtting zone

Overﬁtting zone

Bias

Generalization error

Optimal capacity

Variance

Capacity

Figure 5.6: As capacity increases (x-axis), bias (dotted) tends to decrease and variance (dashed) tends to increase, yielding another U-shaped curve for generalization error (bold curve). If we vary capacity along one axis, there is an optimal capacity, with underﬁtting when the capacity is below this optimum and overﬁtting when it is above. This relationship is similar to the relationship between capacity, underﬁtting, and overﬁtting, discussed in Sec. 5.2 and Fig. 5.3.

eralization error is measured by the MSE (where bias and variance are meaningful components of generalization error), increasing capacity tends to increase variance and decrease bias. This is illustrated in Fig. 5.6, where we see again the U-shaped curve of generalization error as a function of capacity.

5.4.5

Consistency

So far we have discussed the properties of various estimators for a training set of ﬁxed size. Usually, we are also concerned with the behavior of an estimator as the amount of training data grows. In particular, we usually wish that, as the number of data points m in our dataset increases, our point estimates converge to the true value of the corresponding parameters. More formally, we would like that p lim θˆm → θ.

m→∞ p

(5.55)

The symbol → means that the convergence is in probability, i.e. for any > 0, P (|θˆm − θ| > ) → 0 as m → ∞. The condition described by Eq. 5.55 is known as consistency. It is sometimes referred to as weak consistency, with strong consistency referring to the almost sure convergence of θˆ to θ. Almost sure 130

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convergence of a sequence of random variables x (1), x (2) , . . . to a value x occurs when p(lim m→∞ x(m) = x) = 1. Consistency ensures that the bias induced by the estimator is assured to diminish as the number of data examples grows. However, the reverse is not true—asymptotic unbiasedness does not imply consistency. For example, consider estimating the mean parameter µ of a normal distribution N (x; µ, σ2 ), with a dataset consisting of m samples: {x (1) , . . . , x(m) } . We could use the ﬁrst sample x(1) of the dataset as an unbiased estimator: θˆ = x (1). In that case, E(ˆθm ) = θ so the estimator is unbiased no matter how many data points are seen. This, of course, implies that the estimate is asymptotically unbiased. However, this is not a consistent estimator as it is not the case that θˆm → θ as m → ∞.

5.5

Maximum Likelihood Estimation

Previously, we have seen some deﬁnitions of common estimators and analyzed their properties. But where did these estimators come from? Rather than guessing that some function might make a good estimator and then analyzing its bias and variance, we would like to have some principle from which we can derive speciﬁc functions that are good estimators for diﬀerent models. The most common such principle is the maximum likelihood principle. Consider a set of m examples X = {x (1) , . . . , x (m)} drawn independently from the true but unknown data generating distribution pdata (x). Let p model (x; θ) be a parametric family of probability distributions over the same space indexed by θ. In other words, p model(x; θ) maps any conﬁguration x to a real number estimating the true probability pdata (x). The maximum likelihood estimator for θ is then deﬁned as θML = arg max p model(X; θ)

(5.56)

θ

= arg max θ

m

p model(x (i); θ)

(5.57)

i=1

This product over many probabilities can be inconvenient for a variety of reasons. For example, it is prone to numerical underﬂow. To obtain a more convenient but equivalent optimization problem, we observe that taking the logarithm of the likelihood does not change its arg max but does conveniently transform a product

131

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into a sum: θ ML = arg max θ

m

log p model (x (i); θ).

(5.58)

i=1

Because the argmax does not change when we rescale the cost function, we can divide by m to obtain a version of the criterion that is expressed as an expectation with respect to the empirical distribution pˆdata deﬁned by the training data: θ ML = arg max Ex∼ˆpdata log p model (x; θ).

(5.59)

θ

One way to interpret maximum likelihood estimation is to view it as minimizing the dissimilarity between the empirical distribution pˆdata deﬁned by the training set and the model distribution, with the degree of dissimilarity between the two measured by the KL divergence. The KL divergence is given by DKL (ˆ pdata pmodel ) = Ex∼ˆp data [log pˆdata(x) − log p model (x)] .

(5.60)

The term on the left is a function only of the data generating process, not the model. This means when we train the model to minimize the KL divergence, we need only minimize (5.61) − E x∼ˆp data [log pmodel (x)] which is of course the same as the maximization in Eq. 5.59.

Minimizing this KL divergence corresponds exactly to minimizing the crossentropy between the distributions. Many authors use the term “cross-entropy” to identify speciﬁcally the negative log-likelihood of a Bernoulli or softmax distribution, but that is a misnomer. Any loss consisting of a negative log-likelihood is a cross entropy between the empirical distribution deﬁned by the training set and the model. For example, mean squared error is the cross-entropy between the empirical distribution and a Gaussian model. We can thus see maximum likelihood as an attempt to make the model distribution match the empirical distribution pˆdata . Ideally, we would like to match the true data generating distribution p data , but we have no direct access to this distribution. While the optimal θ is the same regardless of whether we are maximizing the likelihood or minimizing the KL divergence, the values of the objective functions are diﬀerent. In software, we often phrase both as minimizing a cost function. Maximum likelihood thus becomes minimization of the negative log-likelihood (NLL), or equivalently, minimization of the cross entropy. The perspective of maximum likelihood as minimum KL divergence becomes helpful in this case because the KL divergence has a known minimum value of zero. The negative log-likelihood can actually become negative when x is real-valued. 132

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5.5.1

Conditional Log-Likelihood and Mean Squared Error

The maximum likelihood estimator can readily be generalized to the case where our goal is to estimate a conditional probability P (y | x ; θ) in order to predict y given x. This is actually the most common situation because it forms the basis for most supervised learning. If X represents all our inputs and Y all our observed targets, then the conditional maximum likelihood estimator is θML = arg max P (Y | X ; θ).

(5.62)

θ

If the examples are assumed to be i.i.d., then this can be decomposed into θML = arg max θ

m i=1

log P (y(i) | x (i); θ).

(5.63)

Example: Linear Regression as Maximum Likelihood Linear regression, introduced earlier in Sec. 5.1.4, may be justiﬁed as a maximum likelihood procedure. Previously, we motivated linear regression as an algorithm that learns to take an input x and produce an output value yˆ. The mapping from x to yˆ is chosen to minimize mean squared error, a criterion that we introduced more or less arbitrarily. We now revisit linear regression from the point of view of maximum likelihood estimation. Instead of producing a single prediction yˆ, we now think of the model as producing a conditional distribution p(y | x). We can imagine that with an inﬁnitely large training set, we might see several training examples with the same input value x but diﬀerent values of y . The goal of the learning algorithm is now to ﬁt the distribution p(y | x) to all of those diﬀerent y values that are all compatible with x. To derive the same linear regression algorithm we obtained before, we deﬁne p(y | x) = N (y; yˆ(x; w), σ2 ). The function yˆ (x; w) gives the prediction of the mean of the Gaussian. In this example, we assume that the variance is ﬁxed to some constant σ 2 chosen by the user. We will see that this choice of the functional form of p(y | x) causes the maximum likelihood estimation procedure to yield the same learning algorithm as we developed before. Since the examples are assumed to be i.i.d., the conditional log-likelihood (Eq. 5.63) is given by m i=1

log p(y (i) | x(i); θ)

(5.64) m

|ˆ m y(i) − y (i)||2 = − m log σ − log(2π ) − 2 2σ 2 i=1 133

(5.65)

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where yˆ(i) is the output of the linear regression on the i-th input x(i) and m is the number of the training examples. Comparing the log-likelihood with the mean squared error, m 1 (i) MSEtrain = (5.66) ||ˆ y − y (i)||2 , m i=1

we immediately see that maximizing the log-likelihood with respect to w yields the same estimate of the parameters w as does minimizing the mean squared error. The two criteria have diﬀerent values but the same location of the optimum. This justiﬁes the use of the MSE as a maximum likelihood estimation procedure. As we will see, the maximum likelihood estimator has several desirable properties.

5.5.2

Properties of Maximum Likelihood

The main appeal of the maximum likelihood estimator is that it can be shown to be the best estimator asymptotically, as the number of examples m → ∞ , in terms of its rate of convergence as m increases. Under appropriate conditions, maximum likelihood estimator has the property of consistency (see Sec. 5.4.5 above), meaning that as the number of training examples approaches inﬁnity, the maximum likelihood estimate of a parameter converges to the true value of the parameter. These conditions are: • The true distribution pdata must lie within the model family p model (·; θ). Otherwise, no estimator can recover pdata . • The true distribution p data must correspond to exactly one value of θ. Otherwise, maximum likelihood can recover the correct p data , but will not be able to determine which value of θ was used by the data generating processing. There are other inductive principles besides the maximum likelihood estimator, many of which share the property of being consistent estimators. However, consistent estimators can diﬀer in their statistic eﬃciency, meaning that one consistent estimator may obtain lower generalization error for a ﬁxed number of samples m, or equivalently, may require fewer examples to obtain a ﬁxed level of generalization error. Statistical eﬃciency is typically studied in the parametric case (like in linear regression) where our goal is to estimate the value of a parameter (and assuming it is possible to identify the true parameter), not the value of a function. A way to measure how close we are to the true parameter is by the expected mean squared error, computing the squared diﬀerence between the estimated and true parameter 134

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values, where the expectation is over m training samples from the data generating distribution. That parametric mean squared error decreases as m increases, and for m large, the Cramér-Rao lower bound (Rao, 1945; Cramér, 1946) shows that no consistent estimator has a lower mean squared error than the maximum likelihood estimator. For these reasons (consistency and eﬃciency), maximum likelihood is often considered the preferred estimator to use for machine learning. When the number of examples is small enough to yield overﬁtting behavior, regularization strategies such as weight decay may be used to obtain a biased version of maximum likelihood that has less variance when training data is limited.

5.6

Bayesian Statistics

So far we have discussed frequentist statistics and approaches based on estimating a single value of θ, then making all predictions thereafter based on that one estimate. Another approach is to consider all possible values of θ when making a prediction. The latter is the domain of Bayesian statistics. As discussed in Sec. 5.4.1, the frequentist perspective is that the true parameter value θ is ﬁxed but unknown, while the point estimate θˆ is a random variable on account of it being a function of the dataset (which is seen as random). The Bayesian perspective on statistics is quite diﬀerent. The Bayesian uses probability to reﬂect degrees of certainty of states of knowledge. The dataset is directly observed and so is not random. On the other hand, the true parameter θ is unknown or uncertain and thus is represented as a random variable. Before observing the data, we represent our knowledge of θ using the prior probability distribution, p(θ ) (sometimes referred to as simply “the prior”). Generally, the machine learning practitioner selects a prior distribution that is quite broad (i.e. with high entropy) to reﬂect a high degree of uncertainty in the value of θ before observing any data. For example, one might assume a priori that θ lies in some ﬁnite range or volume, with a uniform distribution. Many priors instead reﬂect a preference for “simpler” solutions (such as smaller magnitude coeﬃcients, or a function that is closer to being constant). Now consider that we have a set of data samples {x(1), . . . , x (m)}. We can recover the eﬀect of data on our belief about θ by combining the data likelihood p(x(1) , . . . , x(m) | θ) with the prior via Bayes’ rule: p(θ | x (1), . . . , x(m) ) =

p(x (1), . . . , x (m) | θ)p(θ ) p(x(1) , . . . , x(m) )

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(5.67)

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In the scenarios where Bayesian estimation is typically used, the prior begins as a relatively uniform or Gaussian distribution with high entropy, and the observation of the data usually causes the posterior to lose entropy and concentrate around a few highly likely values of the parameters. Relative to maximum likelihood estimation, Bayesian estimation oﬀers two important diﬀerences. First, unlike the maximum likelihood approach that makes predictions using a point estimate of θ, the Bayesian approach is to make predictions using a full distribution over θ. For example, after observing m examples, the predicted distribution over the next data sample, x(m+1) , is given by (m+1) (1) (m) p(x p(x(m+1) | θ)p(θ | x(1) , . . . , x(m) ) dθ. (5.68) | x ,...,x ) = Here each value of θ with positive probability density contributes to the prediction of the next example, with the contribution weighted by the posterior density itself. After having observed {x(1), . . . , x(m)}, if we are still quite uncertain about the value of θ, then this uncertainty is incorporated directly into any predictions we might make. In Sec. 5.4, we discussed how the frequentist approach addresses the uncertainty in a given point estimate of θ by evaluating its variance. The variance of the estimator is an assessment of how the estimate might change with alternative samplings of the observed data. The Bayesian answer to the question of how to deal with the uncertainty in the estimator is to simply integrate over it, which tends to protect well against overﬁtting. This integral is of course just an application of the laws of probability, making the Bayesian approach simple to justify, while the frequentist machinery for constructing an estimator is based on the rather ad hoc decision to summarize all knowledge contained in the dataset with a single point estimate. The second important diﬀerence between the Bayesian approach to estimation and the maximum likelihood approach is due to the contribution of the Bayesian prior distribution. The prior has an inﬂuence by shifting probability mass density towards regions of the parameter space that are preferred a priori. In practice, the prior often expresses a preference for models that are simpler or more smooth. Critics of the Bayesian approach identify the prior as a source of subjective human judgment impacting the predictions. Bayesian methods typically generalize much better when limited training data is available, but typically suﬀer from high computational cost when the number of training examples is large. 136

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Example: Bayesian Linear Regression Here we consider the Bayesian estimation approach to learning the linear regression parameters. In linear regression, we learn a linear mapping from an input vector x ∈ R n to predict the value of a scalar y ∈ R. The prediction is parametrized by the vector w ∈ Rn : yˆ = w x.

(5.69)

Given a set of m training samples (X (train) , y (train) ), we can express the prediction of y over the entire training set as: yˆ (train) = X (train)w.

(5.70)

Expressed as a Gaussian conditional distribution on y(train) , we have p(y (train) | X (train), w) = N (y (train); X (train)w, I ) (5.71) 1 (train) ∝ exp − (y − X (train) w) (y (train) − X (train) w) , 2 (5.72) where we follow the standard MSE formulation in assuming that the Gaussian variance on y is one. In what follows, to reduce the notational burden, we refer to (X (train), y (train)) as simply (X , y ). To determine the posterior distribution over the model parameter vector w , we ﬁrst need to specify a prior distribution. The prior should reﬂect our naive belief about the value of these parameters. While it is sometimes diﬃcult or unnatural to express our prior beliefs in terms of the parameters of the model, in practice we typically assume a fairly broad distribution expressing a high degree of uncertainty about θ. For real-valued parameters it is common to use a Gaussian as a prior distribution: 1 (5.73) p(w) = N (w; µ0, Λ 0) ∝ exp − (w − µ 0) Λ−1 0 (w − µ0 ) 2 where µ0 and Λ 0 are the prior distribution mean vector and covariance matrix respectively.1 With the prior thus speciﬁed, we can now proceed in determining the posterior distribution over the model parameters. p(w | X , y) ∝ p(y | X , w)p(w)

(5.74)

1

Unless there is a reason to assume a particular covariance structure, we typically assume a diagonal covariance matrix Λ0 = diag(λ 0). 137

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1 1 ∝ exp − (y − Xw) (y − Xw) exp − (w − µ 0)Λ −1 0 (w − µ 0 ) 2 2 (5.75) 1 −1 −1 . ∝ exp − −2y Xw + w X Xw + w Λ0 w − 2µ0 Λ0 w 2 (5.76) −1 −1 We now deﬁne Λm = XX + Λ−1 and µ = Λ X y + Λ µ m m 0 . Using 0 0 these new variables, we ﬁnd that the posterior may be rewritten as a Gaussian distribution: 1 1 −1 −1 p(w | X , y) ∝ exp − (w − µ m ) Λm (w − µ m ) + µ mΛm µ m (5.77) 2 2 1 −1 (5.78) ∝ exp − (w − µ m ) Λm (w − µ m ) . 2 All terms that do not include the parameter vector w have been omitted; they are implied by the fact that the distribution must be normalized to integrate to 1. Eq. 3.23 shows how to normalize a multivariate Gaussian distribution. Examining this posterior distribution allows us to gain some intuition for the eﬀect of Bayesian inference. In most situations, we set µ0 to 0. If we set Λ0 = 1α I, then µm gives the same estimate of w as does frequentist linear regression with a weight decay penalty of αw w. One diﬀerence is that the Bayesian estimate is undeﬁned if α is set to zero—-we are not allowed to begin the Bayesian learning process with an inﬁnitely wide prior on w. The more important diﬀerence is that the Bayesian estimate provides a covariance matrix, showing how likely all the diﬀerent values of w are, rather than providing only the estimate µm.

5.6.1

Maximum A Posteriori (MAP) Estimation

While the most principled approach is to make predictions using the full Bayesian posterior distribution over the parameter θ, it is still often desirable to have a single point estimate. One common reason for desiring a point estimate is that most operations involving the Bayesian posterior for most interesting models are intractable, and a point estimate oﬀers a tractable approximation. Rather than simply returning to the maximum likelihood estimate, we can still gain some of the beneﬁt of the Bayesian approach by allowing the prior to inﬂuence the choice of the point estimate. One rational way to do this is to choose the maximum a posteriori (MAP) point estimate. The MAP estimate chooses the point of maximal 138

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posterior probability (or maximal probability density in the more common case of continuous θ): θ MAP = arg max p(θ | x) = arg max log p(x | θ) + log p(θ ). θ

(5.79)

θ

We recognize, above on the right hand side, log p(x | θ), i.e. the standard loglikelihood term, and log p(θ), corresponding to the prior distribution. As an example, consider a linear regression model with a Gaussian prior on the weights w. If this prior is given by N (w; 0, λ1 I2 ), then the log-prior term in Eq. 5.79 is proportional to the familiar λw w weight decay penalty, plus a term that does not depend on w and does not aﬀect the learning process. MAP Bayesian inference with a Gaussian prior on the weights thus corresponds to weight decay. As with full Bayesian inference, MAP Bayesian inference has the advantage of leveraging information that is brought by the prior and cannot be found in the training data. This additional information helps to reduce the variance in the MAP point estimate (in comparison to the ML estimate). However, it does so at the price of increased bias. Many regularized estimation strategies, such as maximum likelihood learning regularized with weight decay, can be interpreted as making the MAP approximation to Bayesian inference. This view applies when the regularization consists of adding an extra term to the objective function that corresponds to log p(θ ). Not all regularization penalties correspond to MAP Bayesian inference. For example, some regularizer terms may not be the logarithm of a probability distribution. Other regularization terms depend on the data, which of course a prior probability distribution is not allowed to do. MAP Bayesian inference provides a straightforward way to design complicated yet interpretable regularization terms. For example, a more complicated penalty term can be derived by using a mixture of Gaussians, rather than a single Gaussian distribution, as the prior (Nowlan and Hinton, 1992).

5.7

Supervised Learning Algorithms

Recall from Sec. 5.1.3 that supervised learning algorithms are, roughly speaking, learning algorithms that learn to associate some input with some output, given a training set of examples of inputs x and outputs y. In many cases the outputs y may be diﬃcult to collect automatically and must be provided by a human “supervisor,” but the term still applies even when the training set targets were collected automatically. 139

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5.7.1

Probabilistic Supervised Learning

Most supervised learning algorithms in this book are based on estimating a probability distribution p(y | x). We can do this simply by using maximum likelihood estimation to ﬁnd the best parameter vector θ for a parametric family of distributions p(y | x; θ). We have already seen that linear regression corresponds to the family p(y | x; θ) = N (y; θ x, I ).

(5.80)

We can generalize linear regression to the classiﬁcation scenario by deﬁning a diﬀerent family of probability distributions. If we have two classes, class 0 and class 1, then we need only specify the probability of one of these classes. The probability of class 1 determines the probability of class 0, because these two values must add up to 1. The normal distribution over real-valued numbers that we used for linear regression is parametrized in terms of a mean. Any value we supply for this mean is valid. A distribution over a binary variable is slightly more complicated, because its mean must always be between 0 and 1. One way to solve this problem is to use the logistic sigmoid function to squash the output of the linear function into the interval (0, 1) and interpret that value as a probability: p(y = 1 | x; θ) = σ (θ x).

(5.81)

This approach is known as logistic regression (a somewhat strange name since we use the model for classiﬁcation rather than regression). In the case of linear regression, we were able to ﬁnd the optimal weights by solving the normal equations. Logistic regression is somewhat more diﬃcult. There is no closed-form solution for its optimal weights. Instead, we must search for them by maximizing the log-likelihood. We can do this by minimizing the negative log-likelihood (NLL) using gradient descent. This same strategy can be applied to essentially any supervised learning problem, by writing down a parametric family of conditional probability distributions over the right kind of input and output variables.

5.7.2

Support Vector Machines

One of the most inﬂuential approaches to supervised learning is the support vector machine (Boser et al., 1992; Cortes and Vapnik, 1995). This model is similar to logistic regression in that it is driven by a linear function w x + b. Unlike logistic 140

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regression, the support vector machine does not provide probabilities, but only outputs a class identity. The SVM predicts that the positive class is present when w x + b is positive. Likewise, it predicts that the negative class is present when w x + b is negative. One key innovation associated with support vector machines is the kernel trick. The kernel trick consists of observing that many machine learning algorithms can be written exclusively in terms of dot products between examples. For example, it can be shown that the linear function used by the support vector machine can be re-written as m w x+b = b+ α ixx (i) (5.82) i=1

where x(i) is a training example and α is a vector of coeﬃcients. Rewriting the learning algorithm this way allows us to replace x by the output of a given feature function φ(x) and the dot product with a function k(x, x(i)) = φ(x)· φ(x(i) ) called a kernel. The · operator represents an inner product analogous to φ(x) φ(x(i)). For some feature spaces, we may not use literally the vector inner product. In some inﬁnite dimensional spaces, we need to use other kinds of inner products, for example, inner products based on integration rather than summation. A complete development of these kinds of inner products is beyond the scope of this book. After replacing dot products with kernel evaluations, we can make predictions using the function f (x ) = b + α ik(x, x(i) ). (5.83) i

This function is nonlinear with respect to x, but the relationship between φ(x) and f (x) is linear. Also, the relationship between α and f(x) is linear. The kernel-based function is exactly equivalent to preprocessing the data by applying φ(x) to all inputs, then learning a linear model in the new transformed space. The kernel trick is powerful for two reasons. First, it allows us to learn models that are nonlinear as a function of x using convex optimization techniques that are guaranteed to converge eﬃciently. This is possible because we consider φ ﬁxed and optimize only α, i.e., the optimization algorithm can view the decision function as being linear in a diﬀerent space. Second, the kernel function k often admits an implementation that is signiﬁcantly more computational eﬃcient than naively constructing two φ(x) vectors and explicitly taking their dot product. In some cases, φ(x) can even be inﬁnite dimensional, which would result in an inﬁnite computational cost for the naive, explicit approach. In many cases, k(x, x ) is a nonlinear, tractable function of x even when φ(x) is intractable. As 141

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an example of an inﬁnite-dimensional feature space with a tractable kernel, we construct a feature mapping φ(x) over the non-negative integers x. Suppose that this mapping returns a vector containing x ones followed by inﬁnitely many zeros. We can write a kernel function k(x, x (i)) = min(x, x(i) ) that is exactly equivalent to the corresponding inﬁnite-dimensional dot product. The most commonly used kernel is the Gaussian kernel k(u, v) = N (u − v ; 0, σ2I)

(5.84)

where N (x; µ, Σ) is the standard normal density. This kernel is also known as the radial basis function (RBF) kernel, because its value decreases along lines in v space radiating outward from u. The Gaussian kernel corresponds to a dot product in an inﬁnite-dimensional space, but the derivation of this space is less straightforward than in our example of the min kernel over the integers. We can think of the Gaussian kernel as performing a kind of template matching. A training example x associated with training label y becomes a template for class y. When a test point x is near x according to Euclidean distance, the Gaussian kernel has a large response, indicating that x is very similar to the x template. The model then puts a large weight on the associated training label y. Overall, the prediction will combine many such training labels weighted by the similarity of the corresponding training examples. Support vector machines are not the only algorithm that can be enhanced using the kernel trick. Many other linear models can be enhanced in this way. The category of algorithms that employ the kernel trick is known as kernel machines or kernel methods (Williams and Rasmussen, 1996; Schölkopf et al., 1999). A major drawback to kernel machines is that the cost of evaluating the decision function is linear in the number of training examples, because the i-th example contributes a term αi k(x, x(i)) to the decision function. Support vector machines are able to mitigate this by learning an α vector that contains mostly zeros. Classifying a new example then requires evaluating the kernel function only for the training examples that have non-zero α i. These training examples are known as support vectors. Kernel machines also suﬀer from a high computational cost of training when the dataset is large. We will revisit this idea in Sec. 5.9. Kernel machines with generic kernels struggle to generalize well. We will explain why in Sec. 5.11. The modern incarnation of deep learning was designed to overcome these limitations of kernel machines. The current deep learning renaissance began when Hinton et al. (2006) demonstrated that a neural network could outperform the RBF kernel SVM on the MNIST benchmark. 142

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5.7.3

Other Simple Supervised Learning Algorithms

We have already brieﬂy encountered another non-probabilistic supervised learning algorithm, nearest neighbor regression. More generally, k-nearest neighbors is a family of techniques that can be used for classiﬁcation or regression. As a non-parametric learning algorithm, k-nearest neighbors is not restricted to a ﬁxed number of parameters. We usually think of the k-nearest neighbors algorithm as not having any parameters, but rather implementing a simple function of the training data. In fact, there is not even really a training stage or learning process. Instead, at test time, when we want to produce an output y for a new test input x, we ﬁnd the k-nearest neighbors to x in the training data X. We then return the average of the corresponding y values in the training set. This works for essentially any kind of supervised learning where we can deﬁne an average over y values. In the case of classiﬁcation, we can average over one-hot code vectors c with c y = 1 and c i = 0 for all other values of i. We can then interpret the average over these one-hot codes as giving a probability distribution over classes. As a non-parametric learning algorithm, k-nearest neighbor can achieve very high capacity. For example, suppose we have a multiclass classiﬁcation task and measure performance with 0-1 loss. In this setting, 1-nearest neighbor converges to double the Bayes error as the number of training examples approaches inﬁnity. The error in excess of the Bayes error results from choosing a single neighbor by breaking ties between equally distant neighbors randomly. When there is inﬁnite training data, all test points x will have inﬁnitely many training set neighbors at distance zero. If we allow the algorithm to use all of these neighbors to vote, rather than randomly choosing one of them, the procedure converges to the Bayes error rate. The high capacity of k-nearest neighbors allows it to obtain high accuracy given a large training set. However, it does so at high computational cost, and it may generalize very badly given a small, ﬁnite training set. One weakness of k-nearest neighbors is that it cannot learn that one feature is more discriminative than another. For example, imagine we have a regression task with x ∈ R100 drawn from an isotropic Gaussian distribution, but only a single variable x1 is relevant to the output. Suppose further that this feature simply encodes the output directly, i.e. that y = x 1 in all cases. Nearest neighbor regression will not be able to detect this simple pattern. The nearest neighbor of most points x will be determined by the large number of features x2 through x 100 , not by the lone feature x1 . Thus the output on small training sets will essentially be random.

143

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Figure 5.7: Diagrams describing how a decision tree works. (Top) Each node of the tree chooses to send the input example to the child node on the left (0) or or the child node on the right (1). Internal nodes are drawn as circles and leaf nodes as squares. Each node is displayed with a binary string identiﬁer corresponding to its position in the tree, obtained by appending a bit to its parent identiﬁer (0=choose left or top, 1=choose right or bottom). (Bottom) The tree divides space into regions. The 2D plane shows how a decision tree might divide R2 . The nodes of the tree are plotted in this plane, with each internal node drawn along the dividing line it uses to categorize examples, and leaf nodes drawn in the center of the region of examples they receive. The result is a piecewise-constant function, with one piece per leaf. Each leaf requires at least one training example to deﬁne, so it is not possible for the decision tree to learn a function that has more local maxima than the number of training examples. 144

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Another type of learning algorithm that also breaks the input space into regions and has separate parameters for each region is the decision tree (Breiman et al., 1984) and its many variants. As shown in Fig. 5.7, each node of the decision tree is associated with a region in the input space, and internal nodes break that region into one sub-region for each child of the node (typically using an axis-aligned cut). Space is thus sub-divided into non-overlapping regions, with a one-to-one correspondence between leaf nodes and input regions. Each leaf node usually maps every point in its input region to the same output. Decision trees are usually trained with specialized algorithms that are beyond the scope of this book. The learning algorithm can be considered non-parametric if it is allowed to learn a tree of arbitrary size, though decision trees are usually regularized with size constraints that turn them into parametric models in practice. Decision trees as they are typically used, with axis-aligned splits and constant outputs within each node, struggle to solve some problems that are easy even for logistic regression. For example, if we have a two-class problem and the positive class occurs wherever x2 > x1 , the decision boundary is not axis-aligned. The decision tree will thus need to approximate the decision boundary with many nodes, implementing a step function that constantly walks back and forth across the true decision function with axis-aligned steps. As we have seen, nearest neighbor predictors and decision trees have many limitations. Nonetheless, they are useful learning algorithms when computational resources are constrained. We can also build intuition for more sophisticated learning algorithms by thinking about the similarities and diﬀerences between sophisticated algorithms and k-NN or decision tree baselines. See Murphy (2012), Bishop (2006), Hastie et al. (2001) or other machine learning textbooks for more material on traditional supervised learning algorithms.

5.8

Unsupervised Learning Algorithms

Recall from Sec. 5.1.3 that unsupervised algorithms are those that experience only “features” but not a supervision signal. The distinction between supervised and unsupervised algorithms is not formally and rigidly deﬁned because there is no objective test for distinguishing whether a value is a feature or a target provided by a supervisor. Informally, unsupervised learning refers to most attempts to extract information from a distribution that do not require human labor to annotate examples. The term is usually associated with density estimation, learning to draw samples from a distribution, learning to denoise data from some distribution, ﬁnding a manifold that the data lies near, or clustering the data into groups of 145

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related examples. A classic unsupervised learning task is to ﬁnd the “best” representation of the data. By ‘best’ we can mean diﬀerent things, but generally speaking we are looking for a representation that preserves as much information about x as possible while obeying some penalty or constraint aimed at keeping the representation simpler or more accessible than x itself. There are multiple ways of deﬁning a simpler representation. Three of the most common include lower dimensional representations, sparse representations and independent representations. Low-dimensional representations attempt to compress as much information about x as possible in a smaller representation. Sparse representations (Barlow, 1989; Olshausen and Field, 1996; Hinton and Ghahramani, 1997) embed the dataset into a representation whose entries are mostly zeroes for most inputs. The use of sparse representations typically requires increasing the dimensionality of the representation, so that the representation becoming mostly zeroes does not discard too much information. This results in an overall structure of the representation that tends to distribute data along the axes of the representation space. Independent representations attempt to disentangle the sources of variation underlying the data distribution such that the dimensions of the representation are statistically independent. Of course these three criteria are certainly not mutually exclusive. Lowdimensional representations often yield elements that have fewer or weaker dependencies than the original high-dimensional data. This is because one way to reduce the size of a representation is to ﬁnd and remove redundancies. Identifying and removing more redundancy allows the dimensionality reduction algorithm to achieve more compression while discarding less information. The notion of representation is one of the central themes of deep learning and therefore one of the central themes in this book. In this section, we develop some simple examples of representation learning algorithms. Together, these example algorithms show how to operationalize all three of the criteria above. Most of the remaining chapters introduce additional representation learning algorithms that develop these criteria in diﬀerent ways or introduce other criteria.

5.8.1

Principal Components Analysis

In Sec. 2.12, we saw that the principal components analysis algorithm provides a means of compressing data. We can also view PCA as an unsupervised learning algorithm that learns a representation of data. This representation is based on two of the criteria for a simple representation described above. PCA learns a 146

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Figure 5.8: PCA learns a linear projection that aligns the direction of greatest variance with the axes of the new space. (Left) The original data consists of samples of x. In this space, the variance might occur along directions that are not axis-aligned. (Right) The transformed data z = x W now varies most along the axis z1. The direction of second most variance is now along z2 .

representation that has lower dimensionality than the original input. It also learns a representation whose elements have no linear correlation with each other. This is a ﬁrst step toward the criterion of learning representations whose elements are statistically independent. To achieve full independence, a representation learning algorithm must also remove the nonlinear relationships between variables. PCA learns an orthogonal, linear transformation of the data that projects an input x to a representation z as shown in Fig. 5.8. In Sec. 2.12, we saw that we could learn a one-dimensional representation that best reconstructs the original data (in the sense of mean squared error) and that this representation actually corresponds to the ﬁrst principal component of the data. Thus we can use PCA as a simple and eﬀective dimensionality reduction method that preserves as much of the information in the data as possible (again, as measured by least-squares reconstruction error). In the following, we will study how the PCA representation decorrelates the original data representation X . Let us consider the m × n-dimensional design matrix X. We will assume that the data has a mean of zero, E[x] = 0. If this is not the case, the data can easily be centered by subtracting the mean from all examples in a preprocessing step. The unbiased sample covariance matrix associated with X is given by: Var[x] =

1 X X. m−1 147

(5.85)

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PCA ﬁnds a representation (through linear transformation) z = xW where Var[z ] is diagonal. In Sec. 2.12, we saw that the principal components of a design matrix X are given by the eigenvectors of X X. From this view, X X = W ΛW .

(5.86)

In this section, we exploit an alternative derivation of the principal components. The principal components may also be obtained via the singular value decomposition. Speciﬁcally, they are the right singular vectors of X. To see this, let W be the right singular vectors in the decomposition X = U ΣW . We then recover the original eigenvector equation with W as the eigenvector basis: X X = U ΣW U ΣW = W Σ2 W . (5.87) The SVD is helpful to show that PCA results in a diagonal Var [z]. Using the SVD of X , we can express the variance of X as: 1 X X m−1 1 = (U ΣW ) U ΣW m−1 1 = W Σ U U ΣW m−1 1 = W Σ 2W , m−1

Var[x] =

(5.88) (5.89) (5.90) (5.91)

where we use the fact that U U = I because the U matrix of the singular value deﬁnition is deﬁned to be orthonormal. This shows that if we take z = xW , we can ensure that the covariance of z is diagonal as required: 1 Z Z m−1 1 = W X XW m−1 1 = W W Σ 2W W m−1 1 = Σ 2, m−1

Var[z ] =

(5.92) (5.93) (5.94) (5.95)

where this time we use the fact that W W = I , again from the deﬁnition of the SVD. 148

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The above analysis shows that when we project the data x to z, via the linear transformation W, the resulting representation has a diagonal covariance matrix (as given by Σ2) which immediately implies that the individual elements of z are mutually uncorrelated. This ability of PCA to transform data into a representation where the elements are mutually uncorrelated is a very important property of PCA. It is a simple example of a representation that attempt to disentangle the unknown factors of variation underlying the data. In the case of PCA, this disentangling takes the form of ﬁnding a rotation of the input space (described by W ) that aligns the principal axes of variance with the basis of the new representation space associated with z. While correlation is an important category of dependency between elements of the data, we are also interested in learning representations that disentangle more complicated forms of feature dependencies. For this, we will need more than what can be done with a simple linear transformation.

5.8.2

k-means Clustering

Another example of a simple representation learning algorithm is k-means clustering. The k-means clustering algorithm divides the training set into k diﬀerent clusters of examples that are near each other. We can thus think of the algorithm as providing a k-dimensional one-hot code vector h representing an input x. If x belongs to cluster i, then h i = 1 and all other entries of the representation h are zero. The one-hot code provided by k-means clustering is an example of a sparse representation, because the majority of its entries are zero for every input. Later, we will develop other algorithms that learn more ﬂexible sparse representations, where more than one entry can be non-zero for each input x. One-hot codes are an extreme example of sparse representations that lose many of the beneﬁts of a distributed representation. The one-hot code still confers some statistical advantages (it naturally conveys the idea that all examples in the same cluster are similar to each other) and it confers the computational advantage that the entire representation may be captured by a single integer. The k-means algorithm works by initializing k diﬀerent centroids {µ (1) , . . . , µ(k)} to diﬀerent values, then alternating between two diﬀerent steps until convergence. In one step, each training example is assigned to cluster i, where i is the index of the nearest centroid µ (i) . In the other step, each centroid µ(i) is updated to the mean of all training examples x (j) assigned to cluster i. 149

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One diﬃculty pertaining to clustering is that the clustering problem is inherently ill-posed, in the sense that there is no single criterion that measures how well a clustering of the data corresponds to the real world. We can measure properties of the clustering such as the average Euclidean distance from a cluster centroid to the members of the cluster. This allows us to tell how well we are able to reconstruct the training data from the cluster assignments. We do not know how well the cluster assignments correspond to properties of the real world. Moreover, there may be many diﬀerent clusterings that all correspond well to some property of the real world. We may hope to ﬁnd a clustering that relates to one feature but obtain a diﬀerent, equally valid clustering that is not relevant to our task. For example, suppose that we run two clustering algorithms on a dataset consisting of images of red trucks, images of red cars, images of gray trucks, and images of gray cars. If we ask each clustering algorithm to ﬁnd two clusters, one algorithm may ﬁnd a cluster of cars and a cluster of trucks, while another may ﬁnd a cluster of red vehicles and a cluster of gray vehicles. Suppose we also run a third clustering algorithm, which is allowed to determine the number of clusters. This may assign the examples to four clusters, red cars, red trucks, gray cars, and gray trucks. This new clustering now at least captures information about both attributes, but it has lost information about similarity. Red cars are in a diﬀerent cluster from gray cars, just as they are in a diﬀerent cluster from gray trucks. The output of the clustering algorithm does not tell us that red cars are more similar to gray cars than they are to gray trucks. They are diﬀerent from both things, and that is all we know. These issues illustrate some of the reasons that we may prefer a distributed representation to a one-hot representation. A distributed representation could have two attributes for each vehicle—one representing its color and one representing whether it is a car or a truck. It is still not entirely clear what the optimal distributed representation is (how can the learning algorithm know whether the two attributes we are interested in are color and car-versus-truck rather than manufacturer and age?) but having many attributes reduces the burden on the algorithm to guess which single attribute we care about, and allows us to measure similarity between objects in a ﬁne-grained way by comparing many attributes instead of just testing whether one attribute matches.

5.9

Stochastic Gradient Descent

Nearly all of deep learning is powered by one very important algorithm: stochastic gradient descent or SGD. Stochastic gradient descent is an extension of the gradient 150

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descent algorithm introduced in Sec. 4.3. A recurring problem in machine learning is that large training sets are necessary for good generalization, but large training sets are also more computationally expensive. The cost function used by a machine learning algorithm often decomposes as a sum over training examples of some per-example loss function. For example, the negative conditional log-likelihood of the training data can be written as m

1 J(θ) = E x,y∼ˆpdata L(x, y, θ) = L(x(i) , y(i) , θ) m i=1

(5.96)

where L is the per-example loss L(x, y, θ) = − log p(y | x; θ ).

For these additive cost functions, gradient descent requires computing m

1 ∇θ J(θ) = ∇θ L(x(i), y(i) , θ). m i=1

(5.97)

The computational cost of this operation is O(m). As the training set size grows to billions of examples, the time to take a single gradient step becomes prohibitively long. The insight of stochastic gradient descent is that the gradient is an expectation. The expectation may be approximately estimated using a small set of samples. Speciﬁcally, on each step of the algorithm, we can sample a minibatch of examples B = {x(1), . . . , x (m ) } drawn uniformly from the training set. The minibatch size m is typically chosen to be a relatively small number of examples, ranging from 1 to a few hundred. Crucially, m is usually held ﬁxed as the training set size m grows. We may ﬁt a training set with billions of examples using updates computed on only a hundred examples. The estimate of the gradient is formed as m

1 g = ∇θ L(x(i) , y(i) , θ). m i=1

(5.98)

using examples from the minibatch B. The stochastic gradient descent algorithm then follows the estimated gradient downhill: θ ← θ − g , where is the learning rate. 151

(5.99)

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Gradient descent in general has often been regarded as slow or unreliable. In the past, the application of gradient descent to non-convex optimization problems was regarded as foolhardy or unprincipled. Today, we know that the machine learning models described in Part II work very well when trained with gradient descent. The optimization algorithm may not be guaranteed to arrive at even a local minimum in a reasonable amount of time, but it often ﬁnds a very low value of the cost function quickly enough to be useful. Stochastic gradient descent has many important uses outside the context of deep learning. It is the main way to train large linear models on very large datasets. For a ﬁxed model size, the cost per SGD update does not depend on the training set size m. In practice, we often use a larger model as the training set size increases, but we are not forced to do so. The number of updates required to reach convergence usually increases with training set size. However, as m approaches inﬁnity, the model will eventually converge to its best possible test error before SGD has sampled every example in the training set. Increasing m further will not extend the amount of training time needed to reach the model’s best possible test error. From this point of view, one can argue that the asymptotic cost of training a model with SGD is O(1) as a function of m. Prior to the advent of deep learning, the main way to learn nonlinear models was to use the kernel trick in combination with a linear model. Many kernel learning algorithms require constructing an m × m matrix Gi,j = k(x(i) , x(j) ). Constructing this matrix has computational cost O(m 2), which is clearly undesirable for datasets with billions of examples. In academia, starting in 2006, deep learning was initially interesting because it was able to generalize to new examples better than competing algorithms when trained on medium-sized datasets with tens of thousands of examples. Soon after, deep learning garnered additional interest in industry, because it provided a scalable way of training nonlinear models on large datasets. Stochastic gradient descent and many enhancements to it are described further in Chapter 8.

5.10

Building a Machine Learning Algorithm

Nearly all deep learning algorithms can be described as particular instances of a fairly simple recipe: combine a speciﬁcation of a dataset, a cost function, an optimization procedure and a model. For example, the linear regression algorithm combines a dataset consisting of 152

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X and y , the cost function J (w, b) = −Ex,y∼ˆp data log pmodel (y | x),

(5.100)

the model speciﬁcation p model (y | x) = N (y; xw + b, 1), and, in most cases, the optimization algorithm deﬁned by solving for where the gradient of the cost is zero using the normal equations. By realizing that we can replace any of these components mostly independently from the others, we can obtain a very wide variety of algorithms. The cost function typically includes at least one term that causes the learning process to perform statistical estimation. The most common cost function is the negative log-likelihood, so that minimizing the cost function causes maximum likelihood estimation. The cost function may also include additional terms, such as regularization terms. For example, we can add weight decay to the linear regression cost function to obtain J (w, b) = λ||w||22 − Ex,y∼p data log pmodel (y | x). (5.101) This still allows closed-form optimization. If we change the model to be nonlinear, then most cost functions can no longer be optimized in closed form. This requires us to choose an iterative numerical optimization procedure, such as gradient descent. The recipe for constructing a learning algorithm by combining models, costs, and optimization algorithms supports both supervised and unsupervised learning. The linear regression example shows how to support supervised learning. Unsupervised learning can be supported by deﬁning a dataset that contains only X and providing an appropriate unsupervised cost and model. For example, we can obtain the ﬁrst PCA vector by specifying that our loss function is J(w) = Ex∼pdata ||x − r(x; w)||22

(5.102)

while our model is deﬁned to have w with norm one and reconstruction function r(x) = w xw. In some cases, the cost function may be a function that we cannot actually evaluate, for computational reasons. In these cases, we can still approximately minimize it using iterative numerical optimization so long as we have some way of approximating its gradients. Most machine learning algorithms make use of this recipe, though it may not immediately be obvious. If a machine learning algorithm seems especially unique or 153

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hand-designed, it can usually be understood as using a special-case optimizer. Some models such as decision trees or k-means require special-case optimizers because their cost functions have ﬂat regions that make them inappropriate for minimization by gradient-based optimizers. Recognizing that most machine learning algorithms can be described using this recipe helps to see the diﬀerent algorithms as part of a taxonomy of methods for doing related tasks that work for similar reasons, rather than as a long list of algorithms that each have separate justiﬁcations.

5.11

Challenges Motivating Deep Learning

The simple machine learning algorithms described in this chapter work very well on a wide variety of important problems. However, they have not succeeded in solving the central problems in AI, such as recognizing speech or recognizing objects. The development of deep learning was motivated in part by the failure of traditional algorithms to generalize well on such AI tasks. This section is about how the challenge of generalizing to new examples becomes exponentially more diﬃcult when working with high-dimensional data, and how the mechanisms used to achieve generalization in traditional machine learning are insuﬃcient to learn complicated functions in high-dimensional spaces. Such spaces also often impose high computational costs. Deep learning was designed to overcome these and other obstacles.

5.11.1

The Curse of Dimensionality

Many machine learning problems become exceedingly diﬃcult when the number of dimensions in the data is high. This phenomenon is known as the curse of dimensionality. Of particular concern is that the number of possible distinct conﬁgurations of a set of variables increases exponentially as the number of variables increases.

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Figure 5.9: As the number of relevant dimensions of the data increases (from left to right), the number of conﬁgurations of interest may grow exponentially. (Left) In this one-dimensional example, we have one variable for which we only care to distinguish 10 regions of interest. With enough examples falling within each of these regions (each region corresponds to a cell in the illustration), learning algorithms can easily generalize correctly. A straightforward way to generalize is to estimate the value of the target function within each region (and possibly interpolate between neighboring regions). (Center) With 2 dimensions (center) it is more diﬃcult to distinguish 10 diﬀerent values of each variable. We need to keep track of up to 10×10=100 regions, and we need at least that many examples to cover all those regions. (Right) With 3 dimensions this grows to 103 = 1000 regions and at least that many examples. For d dimensions and v values to be distinguished along each axis, we seem to need O(v d ) regions and examples. This is an instance of the curse of dimensionality. Figure graciously provided by Nicolas Chapados.

The curse of dimensionality arises in many places in computer science, and especially so in machine learning. One challenge posed by the curse of dimensionality is a statistical challenge. As illustrated in Fig. 5.9, a statistical challenge arises because the number of possible conﬁgurations of x is much larger than the number of training examples. To understand the issue, let us consider that the input space is organized into a grid, like in the ﬁgure. In low dimensions we can describe this space with a low number of grid cells that are mostly occupied by the data. When generalizing to a new data point, we can usually tell what to do simply by inspecting the training examples that lie in the same cell as the new input. For example, if estimating the probability density at some point x, we can just return the number of training examples in the same unit volume cell as x, divided by the total number of training examples. If we wish to classify an example, we can return the most common class of training examples in the same cell. If we are doing regression we can average the target values observed over the examples in that cell. But what about the cells for which we have seen no example? Because in high-dimensional spaces the number of conﬁgurations is going to be huge, much larger than our number of examples, most conﬁgurations will have no training example associated with it. 155

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How could we possibly say something meaningful about these new conﬁgurations? Many traditional machine learning algorithms simply assume that the output at a new point should be approximately the same as the output at the nearest training point.

5.11.2

Local Constancy and Smoothness Regularization

In order to generalize well, machine learning algorithms need to be guided by prior beliefs about what kind of function they should learn. Previously, we have seen these priors incorporated as explicit beliefs in the form of probability distributions over parameters of the model. More informally, we may also discuss prior beliefs as directly inﬂuencing the function itself and only indirectly acting on the parameters via their eﬀect on the function. Additionally, we informally discuss prior beliefs as being expressed implicitly, by choosing algorithms that are biased toward choosing some class of functions over another, even though these biases may not be expressed (or even possible to express) in terms of a probability distribution representing our degree of belief in various functions. Among the most widely used of these implicit “priors” is the smoothness prior or local constancy prior. This prior states that the function we learn should not change very much within a small region. Many simpler algorithms rely exclusively on this prior to generalize well, and as a result they fail to scale to the statistical challenges involved in solving AIlevel tasks. Throughout this book, we will describe how deep learning introduces additional (explicit and implicit) priors in order to reduce the generalization error on sophisticated tasks. Here, we explain why the smoothness prior alone is insuﬃcient for these tasks. There are many diﬀerent ways to implicitly or explicitly express a prior belief that the learned function should be smooth or locally constant. All of these diﬀerent methods are designed to encourage the learning process to learn a function f ∗ that satisﬁes the condition f∗ (x) ≈ f ∗(x + ) (5.103) for most conﬁgurations x and small change . In other words, if we know a good answer for an input x (for example, if x is a labeled training example) then that answer is probably good in the neighborhood of x. If we have several good answers in some neighborhood we would combine them (by some form of averaging or interpolation) to produce an answer that agrees with as many of them as much as possible. An extreme example of the local constancy approach is the k-nearest neighbors 156

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family of learning algorithms. These predictors are literally constant over each region containing all the points x that have the same set of k nearest neighbors in the training set. For k = 1, the number of distinguishable regions cannot be more than the number of training examples. While the k-nearest neighbors algorithm copies the output from nearby training examples, most kernel machines interpolate between training set outputs associated with nearby training examples. An important class of kernels is the family of local kernels where k(u, v) is large when u = v and decreases as u and v grow farther apart from each other. A local kernel can be thought of as a similarity function that performs template matching, by measuring how closely a test example x resembles each training example x (i). Much of the modern motivation for deep learning is derived from studying the limitations of local template matching and how deep models are able to succeed in cases where local template matching fails (Bengio et al., 2006b). Decision trees also suﬀer from the limitations of exclusively smoothness-based learning because they break the input space into as many regions as there are leaves and use a separate parameter (or sometimes many parameters for extensions of decision trees) in each region. If the target function requires a tree with at least n leaves to be represented accurately, then at least n training examples are required to ﬁt the tree. A multiple of n is needed to achieve some level of statistical conﬁdence in the predicted output. In general, to distinguish O (k) regions in input space, all of these methods require O(k) examples. Typically there are O (k) parameters, with O (1) parameters associated with each of the O(k) regions. The case of a nearest neighbor scenario, where each training example can be used to deﬁne at most one region, is illustrated in Fig. 5.10. Is there a way to represent a complex function that has many more regions to be distinguished than the number of training examples? Clearly, assuming only smoothness of the underlying function will not allow a learner to do that. For example, imagine that the target function is a kind of checkerboard. A checkerboard contains many variations but there is a simple structure to them. Imagine what happens when the number of training examples is substantially smaller than the number of black and white squares on the checkerboard. Based on only local generalization and the smoothness or local constancy prior, we would be guaranteed to correctly guess the color of a new point if it lies within the same checkerboard square as a training example. There is no guarantee that the learner could correctly extend the checkerboard pattern to points lying in squares that do not contain training examples. With this prior alone, the only information that an 157

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Figure 5.10: Illustration of how the nearest neighbor algorithm breaks up the input space into regions. An example (represented here by a circle) within each region deﬁnes the region boundary (represented here by the lines). The y value associated with each example deﬁnes what the output should be for all points within the corresponding region. The regions deﬁned by nearest neighbor matching form a geometric pattern called a Voronoi diagram. The number of these contiguous regions cannot grow faster than the number of training examples. While this ﬁgure illustrates the behavior of the nearest neighbor algorithm speciﬁcally, other machine learning algorithms that rely exclusively on the local smoothness prior for generalization exhibit similar behaviors: each training example only informs the learner about how to generalize in some neighborhood immediately surrounding that example.

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example tells us is the color of its square, and the only way to get the colors of the entire checkerboard right is to cover each of its cells with at least one example. The smoothness assumption and the associated non-parametric learning algorithms work extremely well so long as there are enough examples for the learning algorithm to observe high points on most peaks and low points on most valleys of the true underlying function to be learned. This is generally true when the function to be learned is smooth enough and varies in few enough dimensions. In high dimensions, even a very smooth function can change smoothly but in a diﬀerent way along each dimension. If the function additionally behaves diﬀerently in diﬀerent regions, it can become extremely complicated to describe with a set of training examples. If the function is complicated (we want to distinguish a huge number of regions compared to the number of examples), is there any hope to generalize well? The answer to both of these questions is yes. The key insight is that a very large number of regions, e.g., O(2k ), can be deﬁned with O(k) examples, so long as we introduce some dependencies between the regions via additional assumptions about the underlying data generating distribution. In this way, we can actually generalize non-locally (Bengio and Monperrus, 2005; Bengio et al., 2006c). Many diﬀerent deep learning algorithms provide implicit or explicit assumptions that are reasonable for a broad range of AI tasks in order to capture these advantages. Other approaches to machine learning often make stronger, task-speciﬁc assumptions. For example, we could easily solve the checkerboard task by providing the assumption that the target function is periodic. Usually we do not include such strong, task-speciﬁc assumptions into neural networks so that they can generalize to a much wider variety of structures. AI tasks have structure that is much too complex to be limited to simple, manually speciﬁed properties such as periodicity, so we want learning algorithms that embody more general-purpose assumptions. The core idea in deep learning is that we assume that the data was generated by the composition of factors or features, potentially at multiple levels in a hierarchy. Many other similarly generic assumptions can further improve deep learning algorithms. These apparently mild assumptions allow an exponential gain in the relationship between the number of examples and the number of regions that can be distinguished. These exponential gains are described more precisely in Sec. 6.4.1, Sec. 15.4, and Sec. 15.5. The exponential advantages conferred by the use of deep, distributed representations counter the exponential challenges posed by the curse of dimensionality.

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5.11.3

Manifold Learning

An important concept underlying many ideas in machine learning is that of a manifold. A manifold is a connected region. Mathematically, it is a set of points, associated with a neighborhood around each point. From any given point, the manifold locally appears to be a Euclidean space. In everyday life, we experience the surface of the world as a 2-D plane, but it is in fact a spherical manifold in 3-D space. The deﬁnition of a neighborhood surrounding each point implies the existence of transformations that can be applied to move on the manifold from one position to a neighboring one. In the example of the world’s surface as a manifold, one can walk north, south, east, or west. Although there is a formal mathematical meaning to the term “manifold,” in machine learning it tends to be used more loosely to designate a connected set of points that can be approximated well by considering only a small number of degrees of freedom, or dimensions, embedded in a higher-dimensional space. Each dimension corresponds to a local direction of variation. See Fig. 5.11 for an example of training data lying near a one-dimensional manifold embedded in twodimensional space. In the context of machine learning, we allow the dimensionality of the manifold to vary from one point to another. This often happens when a manifold intersects itself. For example, a ﬁgure eight is a manifold that has a single dimension in most places but two dimensions at the intersection at the center. 2.5 2.0 1.5 1.0 0.5 0.0 −0.5 −1.0 0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Figure 5.11: Data sampled from a distribution in a two-dimensional space that is actually concentrated near a one-dimensional manifold, like a twisted string. The solid line indicates the underlying manifold that the learner should infer.

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Many machine learning problems seem hopeless if we expect the machine learning algorithm to learn functions with interesting variations across all of Rn . Manifold learning algorithms surmount this obstacle by assuming that most of R n consists of invalid inputs, and that interesting inputs occur only along a collection of manifolds containing a small subset of points, with interesting variations in the output of the learned function occurring only along directions that lie on the manifold, or with interesting variations happening only when we move from one manifold to another. Manifold learning was introduced in the case of continuous-valued data and the unsupervised learning setting, although this probability concentration idea can be generalized to both discrete data and the supervised learning setting: the key assumption remains that probability mass is highly concentrated. The assumption that the data lies along a low-dimensional manifold may not always be correct or useful. We argue that in the context of AI tasks, such as those that involve processing images, sounds, or text, the manifold assumption is at least approximately correct. The evidence in favor of this assumption consists of two categories of observations. The ﬁrst observation in favor of the manifold hypothesis is that the probability distribution over images, text strings, and sounds that occur in real life is highly concentrated. Uniform noise essentially never resembles structured inputs from these domains. Fig. 5.12 shows how, instead, uniformly sampled points look like the patterns of static that appear on analog television sets when no signal is available. Similarly, if you generate a document by picking letters uniformly at random, what is the probability that you will get a meaningful English-language text? Almost zero, again, because most of the long sequences of letters do not correspond to a natural language sequence: the distribution of natural language sequences occupies a very small volume in the total space of sequences of letters.

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Figure 5.12: Sampling images uniformly at random (by randomly picking each pixel according to a uniform distribution) gives rise to noisy images. Although there is a nonzero probability to generate an image of a face or any other object frequently encountered in AI applications, we never actually observe this happening in practice. This suggests that the images encountered in AI applications occupy a negligible proportion of the volume of image space.

Of course, concentrated probability distributions are not suﬃcient to show that the data lies on a reasonably small number of manifolds. We must also establish that the examples we encounter are connected to each other by other 162

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examples, with each example surrounded by other highly similar examples that may be reached by applying transformations to traverse the manifold. The second argument in favor of the manifold hypothesis is that we can also imagine such neighborhoods and transformations, at least informally. In the case of images, we can certainly think of many possible transformations that allow us to trace out a manifold in image space: we can gradually dim or brighten the lights, gradually move or rotate objects in the image, gradually alter the colors on the surfaces of objects, etc. It remains likely that there are multiple manifolds involved in most applications. For example, the manifold of images of human faces may not be connected to the manifold of images of cat faces. These thought experiments supporting the manifold hypotheses convey some intuitive reasons supporting it. More rigorous experiments (Cayton, 2005; Narayanan and Mitter, 2010; Schölkopf et al., 1998; Roweis and Saul, 2000; Tenenbaum et al., 2000; Brand, 2003; Belkin and Niyogi, 2003; Donoho and Grimes, 2003; Weinberger and Saul, 2004) clearly support the hypothesis for a large class of datasets of interest in AI. When the data lies on a low-dimensional manifold, it can be most natural for machine learning algorithms to represent the data in terms of coordinates on the manifold, rather than in terms of coordinates in R n. In everyday life, we can think of roads as 1-D manifolds embedded in 3-D space. We give directions to speciﬁc addresses in terms of address numbers along these 1-D roads, not in terms of coordinates in 3-D space. Extracting these manifold coordinates is challenging, but holds the promise to improve many machine learning algorithms. This general principle is applied in many contexts. Fig. 5.13 shows the manifold structure of a dataset consisting of faces. By the end of this book, we will have developed the methods necessary to learn such a manifold structure. In Fig. 20.6, we will see how a machine learning algorithm can successfully accomplish this goal. This concludes Part I, which has provided the basic concepts in mathematics and machine learning which are employed throughout the remaining parts of the book. You are now prepared to embark upon your study of deep learning.

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Figure 5.13: Training examples from the QMUL Multiview Face Dataset (Gong et al., 2000) for which the subjects were asked to move in such a way as to cover the two-dimensional manifold corresponding to two angles of rotation. We would like learning algorithms to be able to discover and disentangle such manifold coordinates. Fig. 20.6 illustrates such a feat.

164

Part II

Deep Networks: Modern Practices

165

This part of the book summarizes the state of modern deep learning as it is used to solve practical applications. Deep learning has a long history and many aspirations. Several approaches have been proposed that have yet to entirely bear fruit. Several ambitious goals have yet to be realized. These less-developed branches of deep learning appear in the ﬁnal part of the book. This part focuses only on those approaches that are essentially working technologies that are already used heavily in industry. Modern deep learning provides a very powerful framework for supervised learning. By adding more layers and more units within a layer, a deep network can represent functions of increasing complexity. Most tasks that consist of mapping an input vector to an output vector, and that are easy for a person to do rapidly, can be accomplished via deep learning, given suﬃciently large models and suﬃciently large datasets of labeled training examples. Other tasks, that can not be described as associating one vector to another, or that are diﬃcult enough that a person would require time to think and reﬂect in order to accomplish the task, remain beyond the scope of deep learning for now. This part of the book describes the core parametric function approximation technology that is behind nearly all modern practical applications of deep learning. We begin by describing the feedforward deep network model that is used to represent these functions. Next, we present advanced techniques for regularization and optimization of such models. Scaling these models to large inputs such as high resolution images or long temporal sequences requires specialization. We introduce the convolutional network for scaling to large images and the recurrent neural network for processing temporal sequences. Finally, we present general guidelines for the practical methodology involved in designing, building, and conﬁguring an application involving deep learning, and review some of the applications of deep learning. These chapters are the most important for a practitioner—someone who wants to begin implementing and using deep learning algorithms to solve real-world problems today.

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Chapter 6

Deep Feedforward Networks Deep feedforward networks, also often called feedforward neural networks, or multilayer perceptrons (MLPs), are the quintessential deep learning models. The goal of a feedforward network is to approximate some function f ∗ . For example, for a classiﬁer, y = f ∗(x) maps an input x to a category y. A feedforward network deﬁnes a mapping y = f (x; θ) and learns the value of the parameters θ that result in the best function approximation. These models are called feedforward because information ﬂows through the function being evaluated from x, through the intermediate computations used to deﬁne f , and ﬁnally to the output y. There are no feedback connections in which outputs of the model are fed back into itself. When feedforward neural networks are extended to include feedback connections, they are called recurrent neural networks, presented in Chapter 10. Feedforward networks are of extreme importance to machine learning practitioners. They form the basis of many important commercial applications. For example, the convolutional networks used for object recognition from photos are a specialized kind of feedforward network. Feedforward networks are a conceptual stepping stone on the path to recurrent networks, which power many natural language applications. Feedforward neural networks are called networks because they are typically represented by composing together many diﬀerent functions. The model is associated with a directed acyclic graph describing how the functions are composed together. For example, we might have three functions f (1) , f (2) , and f (3) connected in a chain, to form f(x) = f (3) (f (2) (f (1) (x))). These chain structures are the most commonly used structures of neural networks. In this case, f (1) is called the ﬁrst layer of the network, f (2) is called the second layer, and so on. The overall length 167

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of the chain gives the depth of the model. It is from this terminology that the name “deep learning” arises. The ﬁnal layer of a feedforward network is called the output layer. During neural network training, we drive f(x) to match f ∗(x). The training data provides us with noisy, approximate examples of f ∗(x) evaluated at diﬀerent training points. Each example x is accompanied by a label y ≈ f ∗(x). The training examples specify directly what the output layer must do at each point x; it must produce a value that is close to y. The behavior of the other layers is not directly speciﬁed by the training data. The learning algorithm must decide how to use those layers to produce the desired output, but the training data does not say what each individual layer should do. Instead, the learning algorithm must decide how to use these layers to best implement an approximation of f ∗ . Because the training data does not show the desired output for each of these layers, these layers are called hidden layers. Finally, these networks are called neural because they are loosely inspired by neuroscience. Each hidden layer of the network is typically vector-valued. The dimensionality of these hidden layers determines the width of the model. Each element of the vector may be interpreted as playing a role analogous to a neuron. Rather than thinking of the layer as representing a single vector-to-vector function, we can also think of the layer as consisting of many units that act in parallel, each representing a vector-to-scalar function. Each unit resembles a neuron in the sense that it receives input from many other units and computes its own activation value. The idea of using many layers of vector-valued representation is drawn from neuroscience. The choice of the functions f (i) (x) used to compute these representations is also loosely guided by neuroscientiﬁc observations about the functions that biological neurons compute. However, modern neural network research is guided by many mathematical and engineering disciplines, and the goal of neural networks is not to perfectly model the brain. It is best to think of feedforward networks as function approximation machines that are designed to achieve statistical generalization, occasionally drawing some insights from what we know about the brain, rather than as models of brain function. One way to understand feedforward networks is to begin with linear models and consider how to overcome their limitations. Linear models, such as logistic regression and linear regression, are appealing because they may be ﬁt eﬃciently and reliably, either in closed form or with convex optimization. Linear models also have the obvious defect that the model capacity is limited to linear functions, so the model cannot understand the interaction between any two input variables. To extend linear models to represent nonlinear functions of x, we can apply the linear model not to x itself but to a transformed input φ(x), where φ is a 168

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nonlinear transformation. Equivalently, we can apply the kernel trick described in Sec. 5.7.2, to obtain a nonlinear learning algorithm based on implicitly applying the φ mapping. We can think of φ as providing a set of features describing x, or as providing a new representation for x. The question is then how to choose the mapping φ. 1. One option is to use a very generic φ, such as the inﬁnite-dimensional φ that is implicitly used by kernel machines based on the RBF kernel. If φ(x) is of high enough dimension, we can always have enough capacity to ﬁt the training set, but generalization to the test set often remains poor. Very generic feature mappings are usually based only on the principle of local smoothness and do not encode enough prior information to solve advanced problems. 2. Another option is to manually engineer φ. Until the advent of deep learning, this was the dominant approach. This approach requires decades of human eﬀort for each separate task, with practitioners specializing in diﬀerent domains such as speech recognition or computer vision, and with little transfer between domains. 3. The strategy of deep learning is to learn φ. In this approach, we have a model y = f (x; θ, w ) = φ(x; θ) w. We now have parameters θ that we use to learn φ from a broad class of functions, and parameters w that map from φ(x) to the desired output. This is an example of a deep feedforward network, with φ deﬁning a hidden layer. This approach is the only one of the three that gives up on the convexity of the training problem, but the beneﬁts outweigh the harms. In this approach, we parametrize the representation as φ(x; θ) and use the optimization algorithm to ﬁnd the θ that corresponds to a good representation. If we wish, this approach can capture the beneﬁt of the ﬁrst approach by being highly generic—we do so by using a very broad family φ(x; θ). This approach can also capture the beneﬁt of the second approach. Human practitioners can encode their knowledge to help generalization by designing families φ(x; θ) that they expect will perform well. The advantage is that the human designer only needs to ﬁnd the right general function family rather than ﬁnding precisely the right function. This general principle of improving models by learning features extends beyond the feedforward networks described in this chapter. It is a recurring theme of deep learning that applies to all of the kinds of models described throughout this book. Feedforward networks are the application of this principle to learning deterministic 169

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mappings from x to y that lack feedback connections. Other models presented later will apply these principles to learning stochastic mappings, learning functions with feedback, and learning probability distributions over a single vector. We begin this chapter with a simple example of a feedforward network. Next, we address each of the design decisions needed to deploy a feedforward network. First, training a feedforward network requires making many of the same design decisions as are necessary for a linear model: choosing the optimizer, the cost function, and the form of the output units. We review these basics of gradient-based learning, then proceed to confront some of the design decisions that are unique to feedforward networks. Feedforward networks have introduced the concept of a hidden layer, and this requires us to choose the activation functions that will be used to compute the hidden layer values. We must also design the architecture of the network, including how many layers the network should contain, how these networks should be connected to each other, and how many units should be in each layer. Learning in deep neural networks requires computing the gradients of complicated functions. We present the back-propagation algorithm and its modern generalizations, which can be used to eﬃciently compute these gradients. Finally, we close with some historical perspective.

6.1

Example: Learning XOR

To make the idea of a feedforward network more concrete, we begin with an example of a fully functioning feedforward network on a very simple task: learning the XOR function. The XOR function (“exclusive or”) is an operation on two binary values, x1 and x2. When exactly one of these binary values is equal to 1, the XOR function returns 1. Otherwise, it returns 0. The XOR function provides the target function y = f ∗(x) that we want to learn. Our model provides a function y = f(x;θ ) and our learning algorithm will adapt the parameters θ to make f as similar as possible to f ∗. In this simple example, we will not be concerned with statistical generalization. We want our network to perform correctly on the four points X = {[0,0] , [0, 1], [1, 0], and [1, 1] }. We will train the network on all four of these points. The only challenge is to ﬁt the training set. We can treat this problem as a regression problem and use a mean squared error loss function. We choose this loss function to simplify the math for this example as much as possible. We will see later that there are other, more appropriate 170

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approaches for modeling binary data. Evaluated on our whole training set, the MSE loss function is J(θ) =

1 ∗ (f (x) − f (x; θ))2 . 4

(6.1)

x∈X

Now we must choose the form of our model, f (x; θ). Suppose that we choose a linear model, with θ consisting of w and b. Our model is deﬁned to be f (x; w, b) = xw + b.

(6.2)

We can minimize J (θ) in closed form with respect to w and b using the normal equations. After solving the normal equations, we obtain w = 0 and b = 12. The linear model simply outputs 0.5 everywhere. Why does this happen? Fig. 6.1 shows how a linear model is not able to represent the XOR function. One way to solve this problem is to use a model that learns a diﬀerent feature space in which a linear model is able to represent the solution. Speciﬁcally, we will introduce a very simple feedforward network with one hidden layer containing two hidden units. See Fig. 6.2 for an illustration of this model. This feedforward network has a vector of hidden units h that are computed by a function f (1) (x; W , c). The values of these hidden units are then used as the input for a second layer. The second layer is the output layer of the network. The output layer is still just a linear regression model, but now it is applied to h rather than to x. The network now contains two functions chained together: h = f (1) (x; W , c) and y = f (2)(h; w, b), with the complete model being f (x; W , c, w, b) = f (2)(f (1) (x)). What function should f (1) compute? Linear models have served us well so far, and it may be tempting to make f (1) be linear as well. Unfortunately, if f (1) were linear, then the feedforward network as a whole would remain a linear function of its input. Ignoring the intercept terms for the moment, suppose f (1) (x) = W x and f (2) (h) = h w. Then f(x) = w W x. We could represent this function as f(x) = x w where w = W w. Clearly, we must use a nonlinear function to describe the features. Most neural networks do so using an aﬃne transformation controlled by learned parameters, followed by a ﬁxed, nonlinear function called an activation function. We use that strategy here, by deﬁning h = g(W x + c), where W provides the weights of a linear transformation and c the biases. Previously, to describe a linear regression model, we used a vector of weights and a scalar bias parameter to describe an 171

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Original x Space

Learned h Space

h2

1

x2

1

0

0 0

1

0

x1

1 h1

2

Figure 6.1: Solving the XOR problem by learning a representation. The bold numbers printed on the plot indicate the value that the learned function must output at each point. (Left) A linear model applied directly to the original input cannot implement the XOR function. When x1 = 0, the model’s output must increase as x 2 increases. When x1 = 1, the model’s output must decrease as x2 increases. A linear model must apply a ﬁxed coeﬃcient w 2 to x2. The linear model therefore cannot use the value of x1 to change the coeﬃcient on x 2 and cannot solve this problem. (Right) In the transformed space represented by the features extracted by a neural network, a linear model can now solve the problem. In our example solution, the two points that must have output 1 have been collapsed into a single point in feature space. In other words, the nonlinear features have mapped both x = [1, 0] and x = [0, 1] to a single point in feature space, h = [1 ,0] . The linear model can now describe the function as increasing in h 1 and decreasing in h2. In this example, the motivation for learning the feature space is only to make the model capacity greater so that it can ﬁt the training set. In more realistic applications, learned representations can also help the model to generalize.

172

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y

y

w

h1

h2

h W

x1

x2

x

Figure 6.2: An example of a feedforward network, drawn in two diﬀerent styles. Speciﬁcally, this is the feedforward network we use to solve the XOR example. It has a single hidden layer containing two units. (Left) In this style, we draw every unit as a node in the graph. This style is very explicit and unambiguous but for networks larger than this example it can consume too much space. (Right) In this style, we draw a node in the graph for each entire vector representing a layer’s activations. This style is much more compact. Sometimes we annotate the edges in this graph with the name of the parameters that describe the relationship between two layers. Here, we indicate that a matrix W describes the mapping from x to h, and a vector w describes the mapping from h to y. We typically omit the intercept parameters associated with each layer when labeling this kind of drawing.

aﬃne transformation from an input vector to an output scalar. Now, we describe an aﬃne transformation from a vector x to a vector h, so an entire vector of bias parameters is needed. The activation function g is typically chosen to be a function that is applied element-wise, with h i = g(x W :,i + ci ). In modern neural networks, the default recommendation is to use the rectiﬁed linear unit or ReLU (Jarrett et al., 2009; Nair and Hinton, 2010; Glorot et al., 2011a) deﬁned by the activation function g(z ) = max{0, z } depicted in Fig. 6.3. We can now specify our complete network as

f (x; W , c, w, b) = w max{0, W x + c} + b. We can now specify a solution to the XOR problem. Let 1 1 W = , 1 1 0 c= , −1 1 , w= −2 173

(6.3)

(6.4) (6.5) (6.6)

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g (z ) = max{0, z}

The Rectiﬁed Linear Activation Function

0 0 z

Figure 6.3: The rectiﬁed linear activation function. This activation function is the default activation function recommended for use with most feedforward neural networks. Applying this function to the output of a linear transformation yields a nonlinear transformation. However, the function remains very close to linear, in the sense that is a piecewise linear function with two linear pieces. Because rectiﬁed linear units are nearly linear, they preserve many of the properties that make linear models easy to optimize with gradientbased methods. They also preserve many of the properties that make linear models generalize well. A common principle throughout computer science is that we can build complicated systems from minimal components. Much as a Turing machine’s memory needs only to be able to store 0 or 1 states, we can build a universal function approximator from rectiﬁed linear functions.

174

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and b = 0. We can now walk through the way that the model processes a batch of inputs. Let X be the design matrix containing all four points in the binary input space, with one example per row: 0 0 0 1 X= (6.7) 1 0 . 1 1 The ﬁrst step in the neural network is to multiply the input matrix by the ﬁrst layer’s weight matrix: 0 0 1 1 XW = (6.8) 1 1 . 2 2 Next, we add the bias vector c, to obtain 0 −1 1 0 1 0 2 1

.

(6.9)

In this space, all of the examples lie along a line with slope 1. As we move along this line, the output needs to begin at 0, then rise to 1, then drop back down to 0. A linear model cannot implement such a function. To ﬁnish computing the value of h for each example, we apply the rectiﬁed linear transformation: 0 0 1 0 (6.10) 1 0 . 2 1 This transformation has changed the relationship between the examples. They no longer lie on a single line. As shown in Fig. 6.1, they now lie in a space where a linear model can solve the problem. We ﬁnish by multiplying by the weight vector w: 0 1 . 1 0

175

(6.11)

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The neural network has obtained the correct answer for every example in the batch. In this example, we simply speciﬁed the solution, then showed that it obtained zero error. In a real situation, there might be billions of model parameters and billions of training examples, so one cannot simply guess the solution as we did here. Instead, a gradient-based optimization algorithm can ﬁnd parameters that produce very little error. The solution we described to the XOR problem is at a global minimum of the loss function, so gradient descent could converge to this point. There are other equivalent solutions to the XOR problem that gradient descent could also ﬁnd. The convergence point of gradient descent depends on the initial values of the parameters. In practice, gradient descent would usually not ﬁnd clean, easily understood, integer-valued solutions like the one we presented here.

6.2

Gradient-Based Learning

Designing and training a neural network is not much diﬀerent from training any other machine learning model with gradient descent. In Sec. 5.10, we described how to build a machine learning algorithm by specifying an optimization procedure, a cost function, and a model family. The largest diﬀerence between the linear models we have seen so far and neural networks is that the nonlinearity of a neural network causes most interesting loss functions to become non-convex. This means that neural networks are usually trained by using iterative, gradient-based optimizers that merely drive the cost function to a very low value, rather than the linear equation solvers used to train linear regression models or the convex optimization algorithms with global convergence guarantees used to train logistic regression or SVMs. Convex optimization converges starting from any initial parameters (in theory—in practice it is very robust but can encounter numerical problems). Stochastic gradient descent applied to non-convex loss functions has no such convergence guarantee, and is sensitive to the values of the initial parameters. For feedforward neural networks, it is important to initialize all weights to small random values. The biases may be initialized to zero or to small positive values. The iterative gradient-based optimization algorithms used to train feedforward networks and almost all other deep models will be described in detail in Chapter 8, with parameter initialization in particular discussed in Sec. 8.4. For the moment, it suﬃces to understand that the training algorithm is almost always based on using the gradient to descend the cost function in one way or another. The speciﬁc algorithms are improvements and reﬁnements on the ideas of gradient descent, introduced in Sec. 4.3, and, 176

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more speciﬁcally, are most often improvements of the stochastic gradient descent algorithm, introduced in Sec. 5.9. We can of course, train models such as linear regression and support vector machines with gradient descent too, and in fact this is common when the training set is extremely large. From this point of view, training a neural network is not much diﬀerent from training any other model. Computing the gradient is slightly more complicated for a neural network, but can still be done eﬃciently and exactly. Sec. 6.5 will describe how to obtain the gradient using the back-propagation algorithm and modern generalizations of the back-propagation algorithm. As with other machine learning models, to apply gradient-based learning we must choose a cost function, and we must choose how to represent the output of the model. We now revisit these design considerations with special emphasis on the neural networks scenario.

6.2.1

Cost Functions

An important aspect of the design of a deep neural network is the choice of the cost function. Fortunately, the cost functions for neural networks are more or less the same as those for other parametric models, such as linear models. In most cases, our parametric model deﬁnes a distribution p( y | x;θ ) and we simply use the principle of maximum likelihood. This means we use the cross-entropy between the training data and the model’s predictions as the cost function. Sometimes, we take a simpler approach, where rather than predicting a complete probability distribution over y, we merely predict some statistic of y conditioned on x. Specialized loss functions allow us to train a predictor of these estimates. The total cost function used to train a neural network will often combine one of the primary cost functions described here with a regularization term. We have already seen some simple examples of regularization applied to linear models in Sec. 5.2.2. The weight decay approach used for linear models is also directly applicable to deep neural networks and is among the most popular regularization strategies. More advanced regularization strategies for neural networks will be described in Chapter 7. 6.2.1.1

Learning Conditional Distributions with Maximum Likelihood

Most modern neural networks are trained using maximum likelihood. This means that the cost function is simply the negative log-likelihood, equivalently described 177

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as the cross-entropy between the training data and the model distribution. This cost function is given by J(θ) = −Ex,y∼pˆdata log p model (y | x).

(6.12)

The speciﬁc form of the cost function changes from model to model, depending on the speciﬁc form of log p model . The expansion of the above equation typically yields some terms that do not depend on the model parameters and may be discarded. For example, as we saw in Sec. 5.5.1, if pmodel (y | x) = N (y; f(x; θ), I), then we recover the mean squared error cost, 1 J (θ) = E x,y∼pˆdata ||y − f (x; θ )||2 + const, 2

(6.13)

up to a scaling factor of 12 and a term that does not depend on θ. The discarded constant is based on the variance of the Gaussian distribution, which in this case we chose not to parametrize. Previously, we saw that the equivalence between maximum likelihood estimation with an output distribution and minimization of mean squared error holds for a linear model, but in fact, the equivalence holds regardless of the f (x; θ ) used to predict the mean of the Gaussian. An advantage of this approach of deriving the cost function from maximum likelihood is that it removes the burden of designing cost functions for each model. Specifying a model p(y | x ) automatically determines a cost function log p(y | x). One recurring theme throughout neural network design is that the gradient of the cost function must be large and predictable enough to serve as a good guide for the learning algorithm. Functions that saturate (become very ﬂat) undermine this objective because they make the gradient become very small. In many cases this happens because the activation functions used to produce the output of the hidden units or the output units saturate. The negative log-likelihood helps to avoid this problem for many models. Many output units involve an exp function that can saturate when its argument is very negative. The log function in the negative log-likelihood cost function undoes the exp of some output units. We will discuss the interaction between the cost function and the choice of output unit in Sec. 6.2.2.

One unusual property of the cross-entropy cost used to perform maximum likelihood estimation is that it usually does not have a minimum value when applied to the models commonly used in practice. For discrete output variables, most models are parametrized in such a way that they cannot represent a probability of zero or one, but can come arbitrarily close to doing so. Logistic regression is an example of such a model. For real-valued output variables, if the model 178

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can control the density of the output distribution (for example, by learning the variance parameter of a Gaussian output distribution) then it becomes possible to assign extremely high density to the correct training set outputs, resulting in cross-entropy approaching negative inﬁnity. Regularization techniques described in Chapter 7 provide several diﬀerent ways of modifying the learning problem so that the model cannot reap unlimited reward in this way. 6.2.1.2

Learning Conditional Statistics

Instead of learning a full probability distribution p(y | x; θ) we often want to learn just one conditional statistic of y given x. For example, we may have a predictor f (x; θ ) that we wish to predict the mean of y. If we use a suﬃciently powerful neural network, we can think of the neural network as being able to represent any function f from a wide class of functions, with this class being limited only by features such as continuity and boundedness rather than by having a speciﬁc parametric form. From this point of view, we can view the cost function as being a functional rather than just a function. A functional is a mapping from functions to real numbers. We can thus think of learning as choosing a function rather than merely choosing a set of parameters. We can design our cost functional to have its minimum occur at some speciﬁc function we desire. For example, we can design the cost functional to have its minimum lie on the function that maps x to the expected value of y given x. Solving an optimization problem with respect to a function requires a mathematical tool called calculus of variations, described in Sec. 19.4.2. It is not necessary to understand calculus of variations to understand the content of this chapter. At the moment, it is only necessary to understand that calculus of variations may be used to derive the following two results. Our ﬁrst result derived using calculus of variations is that solving the optimization problem f∗ = arg min Ex,y∼p data||y − f (x)||2 (6.14) f

yields f ∗(x) = Ey∼pdata(y|x) [y],

(6.15)

so long as this function lies within the class we optimize over. In other words, if we could train on inﬁnitely many samples from the true data-generating distribution, minimizing the mean squared error cost function gives a function that predicts the mean of y for each value of x. 179

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Diﬀerent cost functions give diﬀerent statistics. A second result derived using calculus of variations is that f∗ = arg min Ex,y∼p data||y − f (x)||1

(6.16)

f

yields a function that predicts the median value of y for each x, so long as such a function may be described by the family of functions we optimize over. This cost function is commonly called mean absolute error. Unfortunately, mean squared error and mean absolute error often lead to poor results when used with gradient-based optimization. Some output units that saturate produce very small gradients when combined with these cost functions. This is one reason that the cross-entropy cost function is more popular than mean squared error or mean absolute error, even when it is not necessary to estimate an entire distribution p(y | x).

6.2.2

Output Units

The choice of cost function is tightly coupled with the choice of output unit. Most of the time, we simply use the cross-entropy between the data distribution and the model distribution. The choice of how to represent the output then determines the form of the cross-entropy function. Any kind of neural network unit that may be used as an output can also be used as a hidden unit. Here, we focus on the use of these units as outputs of the model, but in principle they can be used internally as well. We revisit these units with additional detail about their use as hidden units in Sec. 6.3. Throughout this section, we suppose that the feedforward network provides a set of hidden features deﬁned by h = f (x; θ). The role of the output layer is then to provide some additional transformation from the features to complete the task that the network must perform. 6.2.2.1

Linear Units for Gaussian Output Distributions

One simple kind of output unit is an output unit based on an aﬃne transformation with no nonlinearity. These are often just called linear units. ˆ = W h+ b. Given features h, a layer of linear output units produces a vector y Linear output layers are often used to produce the mean of a conditional Gaussian distribution: p(y | x) = N (y; yˆ, I ). (6.17) 180

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Maximizing the log-likelihood is then equivalent to minimizing the mean squared error. The maximum likelihood framework makes it straightforward to learn the covariance of the Gaussian too, or to make the covariance of the Gaussian be a function of the input. However, the covariance must be constrained to be a positive deﬁnite matrix for all inputs. It is diﬃcult to satisfy such constraints with a linear output layer, so typically other output units are used to parametrize the covariance. Approaches to modeling the covariance are described shortly, in Sec. 6.2.2.4. Because linear units do not saturate, they pose little diﬃculty for gradientbased optimization algorithms and may be used with a wide variety of optimization algorithms. 6.2.2.2

Sigmoid Units for Bernoulli Output Distributions

Many tasks require predicting the value of a binary variable y . Classiﬁcation problems with two classes can be cast in this form. The maximum-likelihood approach is to deﬁne a Bernoulli distribution over y conditioned on x. A Bernoulli distribution is deﬁned by just a single number. The neural net needs to predict only P (y = 1 | x). For this number to be a valid probability, it must lie in the interval [0, 1]. Satisfying this constraint requires some careful design eﬀort. Suppose we were to use a linear unit, and threshold its value to obtain a valid probability: . (6.18) P (y = 1 | x) = max 0, min 1, w h + b

This would indeed deﬁne a valid conditional distribution, but we would not be able to train it very eﬀectively with gradient descent. Any time that wh + b strayed outside the unit interval, the gradient of the output of the model with respect to its parameters would be 0. A gradient of 0 is typically problematic because the learning algorithm no longer has a guide for how to improve the corresponding parameters. Instead, it is better to use a diﬀerent approach that ensures there is always a strong gradient whenever the model has the wrong answer. This approach is based on using sigmoid output units combined with maximum likelihood. A sigmoid output unit is deﬁned by yˆ = σ w h + b 181

(6.19)

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where σ is the logistic sigmoid function described in Sec. 3.10. We can think of the sigmoid output unit as having two components. First, it uses a linear layer to compute z = w h + b. Next, it uses the sigmoid activation function to convert z into a probability. We omit the dependence on x for the moment to discuss how to deﬁne a probability distribution over y using the value z. The sigmoid can be motivated by constructing an unnormalized probability distribution P˜ (y), which does not sum to 1. We can then divide by an appropriate constant to obtain a valid probability distribution. If we begin with the assumption that the unnormalized log probabilities are linear in y and z, we can exponentiate to obtain the unnormalized probabilities. We then normalize to see that this yields a Bernoulli distribution controlled by a sigmoidal transformation of z : log P˜(y ) = yz P˜(y ) = exp(yz ) P (y ) = 1

exp(yz )

y =0

exp(y z)

P (y ) = σ ((2y − 1)z ) .

(6.20) (6.21) (6.22) (6.23)

Probability distributions based on exponentiation and normalization are common throughout the statistical modeling literature. The z variable deﬁning such a distribution over binary variables is called a logit. This approach to predicting the probabilities in log-space is natural to use with maximum likelihood learning. Because the cost function used with maximum likelihood is − log P (y | x), the log in the cost function undoes the exp of the sigmoid. Without this eﬀect, the saturation of the sigmoid could prevent gradientbased learning from making good progress. The loss function for maximum likelihood learning of a Bernoulli parametrized by a sigmoid is J (θ) = − log P (y | x)

= − log σ ((2y − 1)z ) = ζ ((1 − 2y )z ) .

(6.24) (6.25) (6.26)

This derivation makes use of some properties from Sec. 3.10. By rewriting the loss in terms of the softplus function, we can see that it saturates only when (1 − 2y )z is very negative. Saturation thus occurs only when the model already has the right answer—when y = 1 and z is very positive, or y = 0 and z is very negative. When z has the wrong sign, the argument to the softplus function, 182

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(1 − 2y )z, may be simpliﬁed to |z|. As |z| becomes large while z has the wrong sign, the softplus function asymptotes toward simply returning its argument |z|. The derivative with respect to z asymptotes to sign(z), so, in the limit of extremely incorrect z, the softplus function does not shrink the gradient at all. This property is very useful because it means that gradient-based learning can act to quickly correct a mistaken z . When we use other loss functions, such as mean squared error, the loss can saturate anytime σ(z) saturates. The sigmoid activation function saturates to 0 when z becomes very negative and saturates to 1 when z becomes very positive. The gradient can shrink too small to be useful for learning whenever this happens, whether the model has the correct answer or the incorrect answer. For this reason, maximum likelihood is almost always the preferred approach to training sigmoid output units. Analytically, the logarithm of the sigmoid is always deﬁned and ﬁnite, because the sigmoid returns values restricted to the open interval (0, 1), rather than using the entire closed interval of valid probabilities [0, 1]. In software implementations, to avoid numerical problems, it is best to write the negative log-likelihood as a function of z, rather than as a function of yˆ = σ(z). If the sigmoid function underﬂows to zero, then taking the logarithm of yˆ yields negative inﬁnity. 6.2.2.3

Softmax Units for Multinoulli Output Distributions

Any time we wish to represent a probability distribution over a discrete variable with n possible values, we may use the softmax function. This can be seen as a generalization of the sigmoid function which was used to represent a probability distribution over a binary variable. Softmax functions are most often used as the output of a classiﬁer, to represent the probability distribution over n diﬀerent classes. More rarely, softmax functions can be used inside the model itself, if we wish the model to choose between one of n diﬀerent options for some internal variable. In the case of binary variables, we wished to produce a single number yˆ = P (y = 1 | x).

(6.27)

Because this number needed to lie between 0 and 1, and because we wanted the logarithm of the number to be well-behaved for gradient-based optimization of the log-likelihood, we chose to instead predict a number z = log P˜ (y = 1 | x). Exponentiating and normalizing gave us a Bernoulli distribution controlled by the sigmoid function. 183

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To generalize to the case of a discrete variable with n values, we now need ˆ with yˆi = P (y = i | x). We require not only that each to produce a vector y, element of yˆi be between 0 and 1, but also that the entire vector sums to 1 so that it represents a valid probability distribution. The same approach that worked for the Bernoulli distribution generalizes to the multinoulli distribution. First, a linear layer predicts unnormalized log probabilities: z = W h + b,

(6.28)

where zi = log P˜(y = i | x). The softmax function can then exponentiate and ˆ Formally, the softmax function is given by normalize z to obtain the desired y. exp(zi ) softmax(z)i = . j exp(z j )

(6.29)

As with the logistic sigmoid, the use of the exp function works very well when training the softmax to output a target value y using maximum log-likelihood. In this case, we wish to maximize log P (y = i; z) = log softmax(z)i. Deﬁning the softmax in terms of exp is natural because the log in the log-likelihood can undo the exp of the softmax: log softmax(z )i = zi − log exp(zj ). (6.30) j

The ﬁrst term of Eq. 6.30 shows that the input zi always has a direct contribution to the cost function. Because this term cannot saturate, we know that learning can proceed, even if the contribution of zi to the second term of Eq. 6.30 becomes very small. When maximizing the log-likelihood, the ﬁrst term encourages zi to be pushed up, while the second term encourages all of z to be pushed down. To gain some intuition for the second term, log j exp(z j ), observe that this term can be roughly approximated by maxj zj . This approximation is based on the idea that exp(zk ) is insigniﬁcant for any zk that is noticeably less than max j zj . The intuition we can gain from this approximation is that the negative log-likelihood cost function always strongly penalizes the most active incorrect prediction. If the correct answer already has the largest input to the softmax, then the −zi term and the log j exp(zj ) ≈ maxj zj = zi terms will roughly cancel. This example will then contribute little to the overall training cost, which will be dominated by other examples that are not yet correctly classiﬁed. So far we have discussed only a single example. Overall, unregularized maximum likelihood will drive the model to learn parameters that drive the softmax to predict 184

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the fraction of counts of each outcome observed in the training set: m j=1 1y(j ) =i,x (j )=x softmax(z(x; θ))i ≈ m . j=1 1x(j ) =x

(6.31)

Because maximum likelihood is a consistent estimator, this is guaranteed to happen so long as the model family is capable of representing the training distribution. In practice, limited model capacity and imperfect optimization will mean that the model is only able to approximate these fractions. Many objective functions other than the log-likelihood do not work as well with the softmax function. Speciﬁcally, objective functions that do not use a log to undo the exp of the softmax fail to learn when the argument to the exp becomes very negative, causing the gradient to vanish. In particular, squared error is a poor loss function for softmax units, and can fail to train the model to change its output, even when the model makes highly conﬁdent incorrect predictions (Bridle, 1990). To understand why these other loss functions can fail, we need to examine the softmax function itself. Like the sigmoid, the softmax activation can saturate. The sigmoid function has a single output that saturates when its input is extremely negative or extremely positive. In the case of the softmax, there are multiple output values. These output values can saturate when the diﬀerences between input values become extreme. When the softmax saturates, many cost functions based on the softmax also saturate, unless they are able to invert the saturating activating function. To see that the softmax function responds to the diﬀerence between its inputs, observe that the softmax output is invariant to adding the same scalar to all of its inputs: softmax(z) = softmax(z + c). (6.32) Using this property, we can derive a numerically stable variant of the softmax: softmax(z) = softmax(z − max zi ). i

(6.33)

The reformulated version allows us to evaluate softmax with only small numerical errors even when z contains extremely large or extremely negative numbers. Examining the numerically stable variant, we see that the softmax function is driven by the amount that its arguments deviate from maxi zi . An output softmax(z )i saturates to 1 when the corresponding input is maximal (zi = maxi zi ) and zi is much greater than all of the other inputs. The output softmax(z)i can also saturate to 0 when zi is not maximal and the maximum is much greater. This is a generalization of the way that sigmoid units saturate, and 185

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can cause similar diﬃculties for learning if the loss function is not designed to compensate for it. The argument z to the softmax function can be produced in two diﬀerent ways. The most common is simply to have an earlier layer of the neural network output every element of z, as described above using the linear layer z = W h + b. While straightforward, this approach actually overparametrizes the distribution. The constraint that the n outputs must sum to 1 means that only n − 1 parameters are necessary; the probability of the n -th value may be obtained by subtracting the ﬁrst n − 1 probabilities from 1. We can thus impose a requirement that one element of z be ﬁxed. For example, we can require that zn = 0. Indeed, this is exactly what the sigmoid unit does. Deﬁning P (y = 1 | x) = σ(z) is equivalent to deﬁning P (y = 1 | x) = softmax(z)1 with a two-dimensional z and z1 = 0. Both the n − 1 argument and the n argument approaches to the softmax can describe the same set of probability distributions, but have diﬀerent learning dynamics. In practice, there is rarely much diﬀerence between using the overparametrized version or the restricted version, and it is simpler to implement the overparametrized version. From a neuroscientiﬁc point of view, it is interesting to think of the softmax as a way to create a form of competition between the units that participate in it: the softmax outputs always sum to 1 so an increase in the value of one unit necessarily corresponds to a decrease in the value of others. This is analogous to the lateral inhibition that is believed to exist between nearby neurons in the cortex. At the extreme (when the diﬀerence between the maximal ai and the others is large in magnitude) it becomes a form of winner-take-all (one of the outputs is nearly 1 and the others are nearly 0). The name “softmax” can be somewhat confusing. The function is more closely related to the argmax function than the max function. The term “soft” derives from the fact that the softmax function is continuous and diﬀerentiable. The argmax function, with its result represented as a one-hot vector, is not continuous or diﬀerentiable. The softmax function thus provides a “softened” version of the argmax. The corresponding soft version of the maximum function is softmax(z) z. It would perhaps be better to call the softmax function “softargmax,” but the current name is an entrenched convention. 6.2.2.4

Other Output Types

The linear, sigmoid, and softmax output units described above are the most common. Neural networks can generalize to almost any kind of output layer that we wish. The principle of maximum likelihood provides a guide for how to design 186

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a good cost function for nearly any kind of output layer. In general, if we deﬁne a conditional distribution p(y | x; θ ), the principle of maximum likelihood suggests we use − log p(y | x; θ) as our cost function.

In general, we can think of the neural network as representing a function f(x; θ). The outputs of this function are not direct predictions of the value y. Instead, f (x;θ) = ω provides the parameters for a distribution over y. Our loss function can then be interpreted as − log p(y; ω(x)).

For example, we may wish to learn the variance of a conditional Gaussian for y , given x. In the simple case, where the variance σ2 is a constant, there is a closed form expression because the maximum likelihood estimator of variance is simply the empirical mean of the squared diﬀerence between observations y and their expected value. A computationally more expensive approach that does not require writing special-case code is to simply include the variance as one of the properties of the distribution p(y | x) that is controlled by ω = f(x;θ). The negative log-likelihood − log p(y; ω(x)) will then provide a cost function with the appropriate terms necessary to make our optimization procedure incrementally learn the variance. In the simple case where the standard deviation does not depend on the input, we can make a new parameter in the network that is copied directly into ω. This new parameter might be σ itself or could be a parameter v representing σ 2 or it could be a parameter β representing σ12 , depending on how we choose to parametrize the distribution. We may wish our model to predict a diﬀerent amount of variance in y for diﬀerent values of x. This is called a heteroscedastic model. In the heteroscedastic case, we simply make the speciﬁcation of the variance be one of the values output by f (x; θ). A typical way to do this is to formulate the Gaussian distribution using precision, rather than variance, as described in Eq. 3.22. In the multivariate case it is most common to use a diagonal precision matrix diag(β).

(6.34)

This formulation works well with gradient descent because the formula for the log-likelihood of the Gaussian distribution parametrized by β involves only multiplication by β i and addition of log βi . The gradient of multiplication, addition, and logarithm operations is well-behaved. By comparison, if we parametrized the output in terms of variance, we would need to use division. The division function becomes arbitrarily steep near zero. While large gradients can help learning, arbitrarily large gradients usually result in instability. If we parametrized the output in terms of standard deviation, the log-likelihood would still involve division, and would also involve squaring. The gradient through the squaring operation can vanish near zero, making it diﬃcult to learn parameters that are squared. 187

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Regardless of whether we use standard deviation, variance, or precision, we must ensure that the covariance matrix of the Gaussian is positive deﬁnite. Because the eigenvalues of the precision matrix are the reciprocals of the eigenvalues of the covariance matrix, this is equivalent to ensuring that the precision matrix is positive deﬁnite. If we use a diagonal matrix, or a scalar times the diagonal matrix, then the only condition we need to enforce on the output of the model is positivity. If we suppose that a is the raw activation of the model used to determine the diagonal precision, we can use the softplus function to obtain a positive precision vector: β = ζ(a). This same strategy applies equally if using variance or standard deviation rather than precision or if using a scalar times identity rather than diagonal matrix. It is rare to learn a covariance or precision matrix with richer structure than diagonal. If the covariance is full and conditional, then a parametrization must be chosen that guarantees positive-deﬁniteness of the predicted covariance matrix. This can be achieved by writing Σ(x) = B (x)B (x) , where B is an unconstrained square matrix. One practical issue if the matrix is full rank is that computing the likelihood is expensive, with a d × d matrix requiring O(d3) computation for the determinant and inverse of Σ(x) (or equivalently, and more commonly done, its eigendecomposition or that of B (x)). We often want to perform multimodal regression, that is, to predict real values that come from a conditional distribution p(y | x) that can have several diﬀerent peaks in y space for the same value of x. In this case, a Gaussian mixture is a natural representation for the output (Jacobs et al., 1991; Bishop, 1994). Neural networks with Gaussian mixtures as their output are often called mixture density networks. A Gaussian mixture output with n components is deﬁned by the conditional probability distribution p(y | x) =

n i=1

p(c = i | x)N (y ; µ(i)(x), Σ (i)(x)).

(6.35)

The neural network must have three outputs: a vector deﬁning p( c = i | x), a matrix providing µ(i)(x) for all i, and a tensor providing Σ(i)(x) for all i. These outputs must satisfy diﬀerent constraints: 1. Mixture components p(c = i | x): these form a multinoulli distribution over the n diﬀerent components associated with latent variable 1 c, and can 1

We consider c to be latent because we do not observe it in the data: given input x and target y , it is not possible to know with certainty which Gaussian component was responsible for y, but we can imagine that y was generated by picking one of them, and make that unobserved choice a random variable. 188

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typically be obtained by a softmax over an n-dimensional vector, to guarantee that these outputs are positive and sum to 1. 2. Means µ(i)(x): these indicate the center or mean associated with the i-th Gaussian component, and are unconstrained (typically with no nonlinearity at all for these output units). If y is a d-vector, then the network must output an n × d matrix containing all n of these d-dimensional vectors. Learning these means with maximum likelihood is slightly more complicated than learning the means of a distribution with only one output mode. We only want to update the mean for the component that actually produced the observation. In practice, we do not know which component produced each observation. The expression for the negative log-likelihood naturally weights each example’s contribution to the loss for each component by the probability that the component produced the example. 3. Covariances Σ (i)(x): these specify the covariance matrix for each component i. As when learning a single Gaussian component, we typically use a diagonal matrix to avoid needing to compute determinants. As with learning the means of the mixture, maximum likelihood is complicated by needing to assign partial responsibility for each point to each mixture component. Gradient descent will automatically follow the correct process if given the correct speciﬁcation of the negative log-likelihood under the mixture model. It has been reported that gradient-based optimization of conditional Gaussian mixtures (on the output of neural networks) can be unreliable, in part because one gets divisions (by the variance) which can be numerically unstable (when some variance gets to be small for a particular example, yielding very large gradients). One solution is to clip gradients (see Sec. 10.11.1) while another is to scale the gradients heuristically (Murray and Larochelle, 2014). Gaussian mixture outputs are particularly eﬀective in generative models of speech (Schuster, 1999) or movements of physical objects (Graves, 2013). The mixture density strategy gives a way for the network to represent multiple output modes and to control the variance of its output, which is crucial for obtaining a high degree of quality in these real-valued domains. An example of a mixture density network is shown in Fig. 6.4. In general, we may wish to continue to model larger vectors y containing more variables, and to impose richer and richer structures on these output variables. For example, we may wish for our neural network to output a sequence of characters that forms a sentence. In these cases, we may continue to use the principle of maximum likelihood applied to our model p(y; ω(x)), but the model we use 189

y

CHAPTER 6. DEEP FEEDFORWARD NETWORKS

x

Figure 6.4: Samples drawn from a neural network with a mixture density output layer. The input x is sampled from a uniform distribution and the output y is sampled from pmodel (y | x ). The neural network is able to learn nonlinear mappings from the input to the parameters of the output distribution. These parameters include the probabilities governing which of three mixture components will generate the output as well as the parameters for each mixture component. Each mixture component is Gaussian with predicted mean and variance. All of these aspects of the output distribution are able to vary with respect to the input x, and to do so in nonlinear ways.

to describe y becomes complex enough to be beyond the scope of this chapter. Chapter 10 describes how to use recurrent neural networks to deﬁne such models over sequences, and Part III describes advanced techniques for modeling arbitrary probability distributions.

6.3

Hidden Units

So far we have focused our discussion on design choices for neural networks that are common to most parametric machine learning models trained with gradientbased optimization. Now we turn to an issue that is unique to feedforward neural networks: how to choose the type of hidden unit to use in the hidden layers of the model. The design of hidden units is an extremely active area of research and does not yet have many deﬁnitive guiding theoretical principles. Rectiﬁed linear units are an excellent default choice of hidden unit. Many other types of hidden units are available. It can be diﬃcult to determine when to use which kind (though rectiﬁed linear units are usually an acceptable choice). We 190

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describe here some of the basic intuitions motivating each type of hidden units. These intuitions can be used to suggest when to try out each of these units. It is usually impossible to predict in advance which will work best. The design process consists of trial and error, intuiting that a kind of hidden unit may work well, and then training a network with that kind of hidden unit and evaluating its performance on a validation set. Some of the hidden units included in this list are not actually diﬀerentiable at all input points. For example, the rectiﬁed linear function g(z) = max{0 , z} is not diﬀerentiable at z = 0. This may seem like it invalidates g for use with a gradientbased learning algorithm. In practice, gradient descent still performs well enough for these models to be used for machine learning tasks. This is in part because neural network training algorithms do not usually arrive at a local minimum of the cost function, but instead merely reduce its value signiﬁcantly, as shown in Fig. 4.3. These ideas will be described further in Chapter 8. Because we do not expect training to actually reach a point where the gradient is 0, it is acceptable for the minima of the cost function to correspond to points with undeﬁned gradient. Hidden units that are not diﬀerentiable are usually non-diﬀerentiable at only a small number of points. In general, a function g(z) has a left derivative deﬁned by the slope of the function immediately to the left of z and a right derivative deﬁned by the slope of the function immediately to the right of z. A function is diﬀerentiable at z only if both the left derivative and the right derivative are deﬁned and equal to each other. The functions used in the context of neural networks usually have deﬁned left derivatives and deﬁned right derivatives. In the case of g(z) = max{0, z}, the left derivative at z = 0 is 0 and the right derivative is 1. Software implementations of neural network training usually return one of the one-sided derivatives rather than reporting that the derivative is undeﬁned or raising an error. This may be heuristically justiﬁed by observing that gradientbased optimization on a digital computer is subject to numerical error anyway. When a function is asked to evaluate g(0), it is very unlikely that the underlying value truly was 0. Instead, it was likely to be some small value that was rounded to 0. In some contexts, more theoretically pleasing justiﬁcations are available, but these usually do not apply to neural network training. The important point is that in practice one can safely disregard the non-diﬀerentiability of the hidden unit activation functions described below. Unless indicated otherwise, most hidden units can be described as accepting a vector of inputs x, computing an aﬃne transformation z = W x + b, and then applying an element-wise nonlinear function g(z). Most hidden units are distinguished from each other only by the choice of the form of the activation function g(z ). 191

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6.3.1

Rectiﬁed Linear Units and Their Generalizations

Rectiﬁed linear units use the activation function g(z ) = max{0, z }.

Rectiﬁed linear units are easy to optimize because they are so similar to linear units. The only diﬀerence between a linear unit and a rectiﬁed linear unit is that a rectiﬁed linear unit outputs zero across half its domain. This makes the derivatives through a rectiﬁed linear unit remain large whenever the unit is active. The gradients are not only large but also consistent. The second derivative of the rectifying operation is 0 almost everywhere, and the derivative of the rectifying operation is 1 everywhere that the unit is active. This means that the gradient direction is far more useful for learning than it would be with activation functions that introduce second-order eﬀects. Rectiﬁed linear units are typically used on top of an aﬃne transformation: h = g(W x + b).

(6.36)

When initializing the parameters of the aﬃne transformation, it can be a good practice to set all elements of b to a small, positive value, such as 0.1. This makes it very likely that the rectiﬁed linear units will be initially active for most inputs in the training set and allow the derivatives to pass through. Several generalizations of rectiﬁed linear units exist. Most of these generalizations perform comparably to rectiﬁed linear units and occasionally perform better. One drawback to rectiﬁed linear units is that they cannot learn via gradientbased methods on examples for which their activation is zero. A variety of generalizations of rectiﬁed linear units guarantee that they receive gradient everywhere. Three generalizations of rectiﬁed linear units are based on using a non-zero slope αi when zi < 0: hi = g(z, α) i = max(0, zi ) + α i min(0, z i). Absolute value rectiﬁcation ﬁxes α i = − 1 to obtain g(z) = |z|. It is used for object recognition from images (Jarrett et al., 2009), where it makes sense to seek features that are invariant under a polarity reversal of the input illumination. Other generalizations of rectiﬁed linear units are more broadly applicable. A leaky ReLU (Maas et al., 2013) ﬁxes α i to a small value like 0.01 while a parametric ReLU or PReLU treats αi as a learnable parameter (He et al., 2015). Maxout units (Goodfellow et al., 2013a) generalize rectiﬁed linear units further. Instead of applying an element-wise function g(z), maxout units divide z into groups of k values. Each maxout unit then outputs the maximum element of one 192

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of these groups: g(z ) i = max zj j∈G (i)

(6.37)

where G (i) is the indices of the inputs for group i, { (i − 1)k + 1, . . . , ik}. This provides a way of learning a piecewise linear function that responds to multiple directions in the input x space. A maxout unit can learn a piecewise linear, convex function with up to k pieces. Maxout units can thus be seen as learning the activation function itself rather than just the relationship between units. With large enough k, a maxout unit can learn to approximate any convex function with arbitrary ﬁdelity. In particular, a maxout layer with two pieces can learn to implement the same function of the input x as a traditional layer using the rectiﬁed linear activation function, absolute value rectiﬁcation function, or the leaky or parametric ReLU, or can learn to implement a totally diﬀerent function altogether. The maxout layer will of course be parametrized diﬀerently from any of these other layer types, so the learning dynamics will be diﬀerent even in the cases where maxout learns to implement the same function of x as one of the other layer types. Each maxout unit is now parametrized by k weight vectors instead of just one, so maxout units typically need more regularization than rectiﬁed linear units. They can work well without regularization if the training set is large and the number of pieces per unit is kept low (Cai et al., 2013). Maxout units have a few other beneﬁts. In some cases, one can gain some statistical and computational advantages by requiring fewer parameters. Speciﬁcally, if the features captured by n diﬀerent linear ﬁlters can be summarized without losing information by taking the max over each group of k features, then the next layer can get by with k times fewer weights. Because each unit is driven by multiple ﬁlters, maxout units have some redundancy that helps them to resist a phenomenon called catastrophic forgetting in which neural networks forget how to perform tasks that they were trained on in the past (Goodfellow et al., 2014a). Rectiﬁed linear units and all of these generalizations of them are based on the principle that models are easier to optimize if their behavior is closer to linear. This same general principle of using linear behavior to obtain easier optimization also applies in other contexts besides deep linear networks. Recurrent networks can learn from sequences and produce a sequence of states and outputs. When training them, one needs to propagate information through several time steps, which is much easier when some linear computations (with some directional derivatives being of magnitude near 1) are involved. One of the best-performing recurrent network 193

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architectures, the LSTM, propagates information through time via summation—a particular straightforward kind of such linear activation. This is discussed further in Sec. 10.10.

6.3.2

Logistic Sigmoid and Hyperbolic Tangent

Prior to the introduction of rectiﬁed linear units, most neural networks used the logistic sigmoid activation function g (z ) = σ (z )

(6.38)

or the hyperbolic tangent activation function g(z ) = tanh(z ).

(6.39)

These activation functions are closely related because tanh(z ) = 2σ(2z ) − 1.

We have already seen sigmoid units as output units, used to predict the probability that a binary variable is 1. Unlike piecewise linear units, sigmoidal units saturate across most of their domain—they saturate to a high value when z is very positive, saturate to a low value when z is very negative, and are only strongly sensitive to their input when z is near 0. The widespread saturation of sigmoidal units can make gradient-based learning very diﬃcult. For this reason, their use as hidden units in feedforward networks is now discouraged. Their use as output units is compatible with the use of gradient-based learning when an appropriate cost function can undo the saturation of the sigmoid in the output layer. When a sigmoidal activation function must be used, the hyperbolic tangent activation function typically performs better than the logistic sigmoid. It resembles the identity function more closely, in the sense that tanh (0) = 0 while σ (0) = 12 . Because tanh is similar to identity near 0, training a deep neural network yˆ = w tanh(U tanh(V x)) resembles training a linear model yˆ = w U V x so long as the activations of the network can be kept small. This makes training the tanh network easier. Sigmoidal activation functions are more common in settings other than feedforward networks. Recurrent networks, many probabilistic models, and some autoencoders have additional requirements that rule out the use of piecewise linear activation functions and make sigmoidal units more appealing despite the drawbacks of saturation.

194

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6.3.3

Other Hidden Units

Many other types of hidden units are possible, but are used less frequently. In general, a wide variety of diﬀerentiable functions perform perfectly well. Many unpublished activation functions perform just as well as the popular ones. To provide a concrete example, the authors tested a feedforward network using h = cos(W x + b) on the MNIST dataset and obtained an error rate of less than 1%, which is competitive with results obtained using more conventional activation functions. During research and development of new techniques, it is common to test many diﬀerent activation functions and ﬁnd that several variations on standard practice perform comparably. This means that usually new hidden unit types are published only if they are clearly demonstrated to provide a signiﬁcant improvement. New hidden unit types that perform roughly comparably to known types are so common as to be uninteresting. It would be impractical to list all of the hidden unit types that have appeared in the literature. We highlight a few especially useful and distinctive ones. One possibility is to not have an activation g(z) at all. One can also think of this as using the identity function as the activation function. We have already seen that a linear unit can be useful as the output of a neural network. It may also be used as a hidden unit. If every layer of the neural network consists of only linear transformations, then the network as a whole will be linear. However, it is acceptable for some layers of the neural network to be purely linear. Consider a neural network layer with n inputs and p outputs, h = g(W x + b). We may replace this with two layers, with one layer using weight matrix U and the other using weight matrix V . If the ﬁrst layer has no activation function, then we have essentially factored the weight matrix of the original layer based on W . The factored approach is to compute h = g(V U x + b). If U produces q outputs, then U and V together contain only (n + p)q parameters, while W contains np parameters. For small q, this can be a considerable saving in parameters. It comes at the cost of constraining the linear transformation to be low-rank, but these low-rank relationships are often suﬃcient. Linear hidden units thus oﬀer an eﬀective way of reducing the number of parameters in a network. Softmax units are another kind of unit that is usually used as an output (as described in Sec. 6.2.2.3) but may sometimes be used as a hidden unit. Softmax units naturally represent a probability distribution over a discrete variable with k possible values, so they may be used as a kind of switch. These kinds of hidden units are usually only used in more advanced architectures that explicitly learn to manipulate memory, described in Sec. 10.12. 195

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A few other reasonably common hidden unit types include: 1 2 • Radial basis function or RBF unit: hi = exp − σ2 ||W:,i − x|| . This i function becomes more active as x approaches a template W:,i . Because it saturates to 0 for most x, it can be diﬃcult to optimize. • Softplus: g(a) = ζ(a) = log(1 + ea ). This is a smooth version of the rectiﬁer, introduced by Dugas et al. (2001) for function approximation and by Nair and Hinton (2010) for the conditional distributions of undirected probabilistic models. Glorot et al. (2011a) compared the softplus and rectiﬁer and found better results with the latter. The use of the softplus is generally discouraged. The softplus demonstrates that the performance of hidden unit types can be very counterintuitive—one might expect it to have an advantage over the rectiﬁer due to being diﬀerentiable everywhere or due to saturating less completely, but empirically it does not. • Hard tanh: this is shaped similarly to the tanh and the rectiﬁer but unlike the latter, it is bounded, g(a) = max (− 1, min(1, a)). It was introduced by Collobert (2004). Hidden unit design remains an active area of research and many useful hidden unit types remain to be discovered.

6.4

Architecture Design

Another key design consideration for neural networks is determining the architecture. The word architecture refers to the overall structure of the network: how many units it should have and how these units should be connected to each other. Most neural networks are organized into groups of units called layers. Most neural network architectures arrange these layers in a chain structure, with each layer being a function of the layer that preceded it. In this structure, the ﬁrst layer is given by (1) (1) (1) (1) h =g W x+b , (6.40) the second layer is given by

and so on.

h(2) = g(2) W (2) h(1) + b(2) , 196

(6.41)

CHAPTER 6. DEEP FEEDFORWARD NETWORKS

In these chain-based architectures, the main architectural considerations are to choose the depth of the network and the width of each layer. As we will see, a network with even one hidden layer is suﬃcient to ﬁt the training set. Deeper networks often are able to use far fewer units per layer and far fewer parameters and often generalize to the test set, but are also often harder to optimize. The ideal network architecture for a task must be found via experimentation guided by monitoring the validation set error.

6.4.1

Universal Approximation Properties and Depth

A linear model, mapping from features to outputs via matrix multiplication, can by deﬁnition represent only linear functions. It has the advantage of being easy to train because many loss functions result in convex optimization problems when applied to linear models. Unfortunately, we often want to learn nonlinear functions. At ﬁrst glance, we might presume that learning a nonlinear function requires designing a specialized model family for the kind of nonlinearity we want to learn. Fortunately, feedforward networks with hidden layers provide a universal approximation framework. Speciﬁcally, the universal approximation theorem (Hornik et al., 1989; Cybenko, 1989) states that a feedforward network with a linear output layer and at least one hidden layer with any “squashing” activation function (such as the logistic sigmoid activation function) can approximate any Borel measurable function from one ﬁnite-dimensional space to another with any desired non-zero amount of error, provided that the network is given enough hidden units. The derivatives of the feedforward network can also approximate the derivatives of the function arbitrarily well (Hornik et al., 1990). The concept of Borel measurability is beyond the scope of this book; for our purposes it suﬃces to say that any continuous function on a closed and bounded subset of Rn is Borel measurable and therefore may be approximated by a neural network. A neural network may also approximate any function mapping from any ﬁnite dimensional discrete space to another. While the original theorems were ﬁrst stated in terms of units with activation functions that saturate both for very negative and for very positive arguments, universal approximation theorems have also been proven for a wider class of activation functions, which includes the now commonly used rectiﬁed linear unit (Leshno et al., 1993). The universal approximation theorem means that regardless of what function we are trying to learn, we know that a large MLP will be able to represent this function. However, we are not guaranteed that the training algorithm will be able to learn that function. Even if the MLP is able to represent the function, learning can fail for two diﬀerent reasons. First, the optimization algorithm used for training 197

CHAPTER 6. DEEP FEEDFORWARD NETWORKS

may not be able to ﬁnd the value of the parameters that corresponds to the desired function. Second, the training algorithm might choose the wrong function due to overﬁtting. Recall from Sec. 5.2.1 that the “no free lunch” theorem shows that there is no universally superior machine learning algorithm. Feedforward networks provide a universal system for representing functions, in the sense that, given a function, there exists a feedforward network that approximates the function. There is no universal procedure for examining a training set of speciﬁc examples and choosing a function that will generalize to points not in the training set. The universal approximation theorem says that there exists a network large enough to achieve any degree of accuracy we desire, but the theorem does not say how large this network will be. Barron (1993) provides some bounds on the size of a single-layer network needed to approximate a broad class of functions. Unfortunately, in the worse case, an exponential number of hidden units (possibly with one hidden unit corresponding to each input conﬁguration that needs to be distinguished) may be required. This is easiest to see in the binary case: the n number of possible binary functions on vectors v ∈ {0, 1}n is 22 and selecting one such function requires 2n bits, which will in general require O(2 n ) degrees of freedom. In summary, a feedforward network with a single layer is suﬃcient to represent any function, but the layer may be infeasibly large and may fail to learn and generalize correctly. In many circumstances, using deeper models can reduce the number of units required to represent the desired function and can reduce the amount of generalization error. There exist families of functions which can be approximated eﬃciently by an architecture with depth greater than some value d, but which require a much larger model if depth is restricted to be less than or equal to d. In many cases, the number of hidden units required by the shallow model is exponential in n. Such results were ﬁrst proven for models that do not resemble the continuous, diﬀerentiable neural networks used for machine learning, but have since been extended to these models. The ﬁrst results were for circuits of logic gates (Håstad, 1986). Later work extended these results to linear threshold units with non-negative weights (Håstad and Goldmann, 1991; Hajnal et al., 1993), and then to networks with continuous-valued activations (Maass, 1992; Maass et al., 1994). Many modern neural networks use rectiﬁed linear units. Leshno et al. (1993) demonstrated that shallow networks with a broad family of non-polynomial activation functions, including rectiﬁed linear units, have universal approximation properties, but these results do not address the questions of depth or eﬃciency—they specify only that a suﬃciently wide rectiﬁer network could represent any function. Pascanu et al. 198

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(2013b) and Montufar et al. (2014) showed that functions representable with a deep rectiﬁer net can require an exponential number of hidden units with a shallow (one hidden layer) network. More precisely, they showed that piecewise linear networks (which can be obtained from rectiﬁer nonlinearities or maxout units) can represent functions with a number of regions that is exponential in the depth of the network. Fig. 6.5 illustrates how a network with absolute value rectiﬁcation creates mirror images of the function computed on top of some hidden unit, with respect to the input of that hidden unit. Each hidden unit speciﬁes where to fold the input space in order to create mirror responses (on both sides of the absolute value nonlinearity). By composing these folding operations, we obtain an exponentially large number of piecewise linear regions which can capture all kinds of regular (e.g., repeating) patterns.

Figure 6.5: An intuitive, geometric explanation of the exponential advantage of deeper rectiﬁer networks formally shown by Pascanu et al. (2014a) and by Montufar et al. (2014). (Left) An absolute value rectiﬁcation unit has the same output for every pair of mirror points in its input. The mirror axis of symmetry is given by the hyperplane deﬁned by the weights and bias of the unit. A function computed on top of that unit (the green decision surface) will be a mirror image of a simpler pattern across that axis of symmetry. (Center) The function can be obtained by folding the space around the axis of symmetry. (Right) Another repeating pattern can be folded on top of the ﬁrst (by another downstream unit) to obtain another symmetry (which is now repeated four times, with two hidden layers).

More precisely, the main theorem in Montufar et al. (2014) states that the number of linear regions carved out by a deep rectiﬁer network with d inputs, depth l, and n units per hidden layer, is n d(l−1) d O n , (6.42) d i.e., exponential in the depth l. In the case of maxout networks with k ﬁlters per unit, the number of linear regions is (l−1)+d O k . (6.43) 199

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Of course, there is no guarantee that the kinds of functions we want to learn in applications of machine learning (and in particular for AI) share such a property. We may also want to choose a deep model for statistical reasons. Any time we choose a speciﬁc machine learning algorithm, we are implicitly stating some set of prior beliefs we have about what kind of function the algorithm should learn. Choosing a deep model encodes a very general belief that the function we want to learn should involve composition of several simpler functions. This can be interpreted from a representation learning point of view as saying that we believe the learning problem consists of discovering a set of underlying factors of variation that can in turn be described in terms of other, simpler underlying factors of variation. Alternately, we can interpret the use of a deep architecture as expressing a belief that the function we want to learn is a computer program consisting of multiple steps, where each step makes use of the previous step’s output. These intermediate outputs are not necessarily factors of variation, but can instead be analogous to counters or pointers that the network uses to organize its internal processing. Empirically, greater depth does seem to result in better generalization for a wide variety of tasks (Bengio et al., 2007; Erhan et al., 2009; Bengio, 2009; Mesnil et al., 2011; Ciresan et al., 2012; Krizhevsky et al., 2012; Sermanet et al., 2013; Farabet et al., 2013; Couprie et al., 2013; Kahou et al., 2013; Goodfellow et al., 2014d; Szegedy et al., 2014a). See Fig. 6.6 and Fig. 6.7 for examples of some of these empirical results. This suggests that using deep architectures does indeed express a useful prior over the space of functions the model learns.

6.4.2

Other Architectural Considerations

So far we have described neural networks as being simple chains of layers, with the main considerations being the depth of the network and the width of each layer. In practice, neural networks show considerably more diversity. Many neural network architectures have been developed for speciﬁc tasks. Specialized architectures for computer vision called convolutional networks are described in Chapter 9. Feedforward networks may also be generalized to the recurrent neural networks for sequence processing, described in Chapter 10, which have their own architectural considerations. In general, the layers need not be connected in a chain, even though this is the most common practice. Many architectures build a main chain but then add extra architectural features to it, such as skip connections going from layer i to layer i + 2 or higher. These skip connections make it easier for the gradient to ﬂow from output layers to layers nearer the input. 200

CHAPTER 6. DEEP FEEDFORWARD NETWORKS

Figure 6.6: Empirical results showing that deeper networks generalize better when used to transcribe multi-digit numbers from photographs of addresses. Data from Goodfellow et al. (2014d). The test set accuracy consistently increases with increasing depth. See Fig. 6.7 for a control experiment demonstrating that other increases to the model size do not yield the same eﬀect.

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Eﬀect of Number of Parameters

Test accuracy (%)

97

3, convolutional 3, fully connected 11, convolutional

96 95 94 93 92 91 0.0

0. 2

0. 4

0.6

Number of parameters

0. 8

1. 0 ×108

Figure 6.7: Deeper models tend to perform better. This is not merely because the model is larger. This experiment from Goodfellow et al. (2014d) shows that increasing the number of parameters in layers of convolutional networks without increasing their depth is not nearly as eﬀective at increasing test set performance. The legend indicates the depth of network used to make each curve and whether the curve represents variation in the size of the convolutional or the fully connected layers. We observe that shallow models in this context overﬁt at around 20 million parameters while deep ones can beneﬁt from having over 60 million. This suggests that using a deep model expresses a useful preference over the space of functions the model can learn. Speciﬁcally, it expresses a belief that the function should consist of many simpler functions composed together. This could result either in learning a representation that is composed in turn of simpler representations (e.g., corners deﬁned in terms of edges) or in learning a program with sequentially dependent steps (e.g., ﬁrst locate a set of objects, then segment them from each other, then recognize them).

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Another key consideration of architecture design is exactly how to connect a pair of layers to each other. In the default neural network layer described by a linear transformation via a matrix W , every input unit is connected to every output unit. Many specialized networks in the chapters ahead have fewer connections, so that each unit in the input layer is connected to only a small subset of units in the output layer. These strategies for reducing the number of connections reduce the number of parameters and the amount of computation required to evaluate the network, but are often highly problem-dependent. For example, convolutional networks, described in Chapter 9, use specialized patterns of sparse connections that are very eﬀective for computer vision problems. In this chapter, it is diﬃcult to give much more speciﬁc advice concerning the architecture of a generic neural network. Subsequent chapters develop the particular architectural strategies that have been found to work well for diﬀerent application domains.

6.5

Back-Propagation and Other Diﬀerentiation Algorithms

When we use a feedforward neural network to accept an input x and produce an ˆ information ﬂows forward through the network. The inputs x provide output y, the initial information that then propagates up to the hidden units at each layer ˆ This is called forward propagation. During training, and ﬁnally produces y. forward propagation can continue onward until it produces a scalar cost J (θ). The back-propagation algorithm (Rumelhart et al., 1986a), often simply called backprop, allows the information from the cost to then ﬂow backwards through the network, in order to compute the gradient. Computing an analytical expression for the gradient is straightforward, but numerically evaluating such an expression can be computationally expensive. The back-propagation algorithm does so using a simple and inexpensive procedure. The term back-propagation is often misunderstood as meaning the whole learning algorithm for multi-layer neural networks. Actually, back-propagation refers only to the method for computing the gradient, while another algorithm, such as stochastic gradient descent, is used to perform learning using this gradient. Furthermore, back-propagation is often misunderstood as being speciﬁc to multilayer neural networks, but in principle it can compute derivatives of any function (for some functions, the correct response is to report that the derivative of the function is undeﬁned). Speciﬁcally, we will describe how to compute the gradient ∇x f( x, y) for an arbitrary function f, where x is a set of variables whose derivatives are desired, and y is an additional set of variables that are inputs to the function 203

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but whose derivatives are not required. In learning algorithms, the gradient we most often require is the gradient of the cost function with respect to the parameters, ∇θ J(θ). Many machine learning tasks involve computing other derivatives, either as part of the learning process, or to analyze the learned model. The backpropagation algorithm can be applied to these tasks as well, and is not restricted to computing the gradient of the cost function with respect to the parameters. The idea of computing derivatives by propagating information through a network is very general, and can be used to compute values such as the Jacobian of a function f with multiple outputs. We restrict our description here to the most commonly used case where f has a single output.

6.5.1

Computational Graphs

So far we have discussed neural networks with a relatively informal graph language. To describe the back-propagation algorithm more precisely, it is helpful to have a more precise computational graph language. Many ways of formalizing computation as graphs are possible. Here, we use each node in the graph to indicate a variable. The variable may be a scalar, vector, matrix, tensor, or even a variable of another type. To formalize our graphs, we also need to introduce the idea of an operation. An operation is a simple function of one or more variables. Our graph language is accompanied by a set of allowable operations. Functions more complicated than the operations in this set may be described by composing many operations together. Without loss of generality, we deﬁne an operation to return only a single output variable. This does not lose generality because the output variable can have multiple entries, such as a vector. Software implementations of back-propagation usually support operations with multiple outputs, but we avoid this case in our description because it introduces many extra details that are not important to conceptual understanding. If a variable y is computed by applying an operation to a variable x, then we draw a directed edge from x to y. We sometimes annotate the output node with the name of the operation applied, and other times omit this label when the operation is clear from context. Examples of computational graphs are shown in Fig. 6.8.

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yˆ σ

z

u(1)

u(2) +

dot

×

y

x

x

w

(a)

(b)

u(2)

H

relu

U (2)

u(3) ×

sum U (1)

b

u(1)

yˆ

+

dot

sqr

matmul X

W

x

b

(c)

w

λ

(d)

Figure 6.8: Examples of computational graphs. (a) The graph using the × operation to compute z = xy. (b) The graph for the logistic regression prediction yˆ = σ x w + b . Some of the intermediate expressions do not have names in the algebraic expression but need names in the graph. We simply name the i-th such variable u(i) . (c) The computational graph for the expression H = max{0, XW + b}, which computes a design matrix of rectiﬁed linear unit activations H given a design matrix containing a minibatch of inputs X. (d) Examples a–c applied at most one operation to each variable, but it is possible to apply more than one operation. Here we show a computation graph that applies more than one operation to the weights w of a linear regression model. The 2 y ˆ λ weights are used to make the both the prediction and the weight decay penalty i wi .

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6.5.2

Chain Rule of Calculus

The chain rule of calculus (not to be confused with the chain rule of probability) is used to compute the derivatives of functions formed by composing other functions whose derivatives are known. Back-propagation is an algorithm that computes the chain rule, with a speciﬁc order of operations that is highly eﬃcient. Let x be a real number, and let f and g both be functions mapping from a real number to a real number. Suppose that y = g(x) and z = f (g(x)) = f (y). Then the chain rule states that dz dz dy = . (6.44) dx dy dx We can generalize this beyond the scalar case. Suppose that x ∈ Rm, y ∈ R n, g maps from R m to Rn , and f maps from R n to R . If y = g(x) and z = f( y), then ∂z ∂yj ∂z = . ∂xi ∂y j ∂x i j

(6.45)

In vector notation, this may be equivalently written as ∇x z = where

∂y ∂x

∂y ∂x

∇y z,

(6.46)

is the n × m Jacobian matrix of g.

From this we see that the gradient of a variable x can be obtained by multiplying a Jacobian matrix ∂y ∂x by a gradient ∇y z. The back-propagation algorithm consists of performing such a Jacobian-gradient product for each operation in the graph. Usually we do not apply the back-propagation algorithm merely to vectors, but rather to tensors of arbitrary dimensionality. Conceptually, this is exactly the same as back-propagation with vectors. The only diﬀerence is how the numbers are arranged in a grid to form a tensor. We could imagine ﬂattening each tensor into a vector before we run back-propagation, computing a vector-valued gradient, and then reshaping the gradient back into a tensor. In this rearranged view, back-propagation is still just multiplying Jacobians by gradients. To denote the gradient of a value z with respect to a tensor X, we write ∇ X z, just as if X were a vector. The indices into X now have multiple coordinates—for example, a 3-D tensor is indexed by three coordinates. We can abstract this away by using a single variable i to represent the complete tuple of indices. For all ∂z possible index tuples i, (∇X z)i gives ∂X . This is exactly the same as how for all i 206

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∂z possible integer indices i into a vector, (∇x z)i gives ∂x . Using this notation, we i can write the chain rule as it applies to tensors. If Y = g (X) and z = f (Y), then

∇X z =

6.5.3

(∇X Yj )

j

∂z . ∂Y j

(6.47)

Recursively Applying the Chain Rule to Obtain Backprop

Using the chain rule, it is straightforward to write down an algebraic expression for the gradient of a scalar with respect to any node in the computational graph that produced that scalar. However, actually evaluating that expression in a computer introduces some extra considerations. Speciﬁcally, many subexpressions may be repeated several times within the overall expression for the gradient. Any procedure that computes the gradient will need to choose whether to store these subexpressions or to recompute them several times. An example of how these repeated subexpressions arise is given in Fig. 6.9. In some cases, computing the same subexpression twice would simply be wasteful. For complicated graphs, there can be exponentially many of these wasted computations, making a naive implementation of the chain rule infeasible. In other cases, computing the same subexpression twice could be a valid way to reduce memory consumption at the cost of higher runtime. We ﬁrst begin by a version of the back-propagation algorithm that speciﬁes the actual gradient computation directly (Algorithm 6.2 along with Algorithm 6.1 for the associated forward computation), in the order it will actually be done and according to the recursive application of chain rule. One could either directly perform these computations or view the description of the algorithm as a symbolic speciﬁcation of the computational graph for computing the back-propagation. However, this formulation does not make explicit the manipulation and the construction of the symbolic graph that performs the gradient computation. Such a formulation is presented below in Sec. 6.5.6, with Algorithm 6.5, where we also generalize to nodes that contain arbitrary tensors. First consider a computational graph describing how to compute a single scalar (say the loss on a training example). This scalar is the quantity whose u gradient we want to obtain, with respect to the ni input nodes u(1) to u (ni ) . In (n) other words we wish to compute ∂u for all i ∈ {1, 2, . . . , ni}. In the application ∂u (i) of back-propagation to computing gradients for gradient descent over parameters, u(n) will be the cost associated with an example or a minibatch, while u(1) to u(n i) correspond to the parameters of the model. (n)

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We will assume that the nodes of the graph have been ordered in such a way that we can compute their output one after the other, starting at u(ni +1) and going up to u(n). As deﬁned in Algorithm 6.1, each node u (i) is associated with an operation f (i) and is computed by evaluating the function u(i) = f (A(i))

(6.48)

where A (i) is the set of all nodes that are parents of u(i) . Algorithm 6.1 A procedure that performs the computations mapping n i inputs u(1) to u(n i) to an output u(n) . This deﬁnes a computational graph where each node computes numerical value u(i) by applying a function f (i) to the set of arguments A(i) that comprises the values of previous nodes u (j), j < i, with j ∈ P a(u(i) ). The input to the computational graph is the vector x, and is set into the ﬁrst ni nodes u(1) to u (ni ) . The output of the computational graph is read oﬀ the last (output) node u(n). for i = 1, . . . , ni do u(i) ← xi end for for i = ni + 1, . . . , n do A(i) ← {u (j) | j ∈ P a(u(i) )} u(i) ← f (i)(A (i)) end for return u(n) That algorithm speciﬁes the forward propagation computation, which we could put in a graph G. In order to perform back-propagation, we can construct a computational graph that depends on G and adds to it an extra set of nodes. These form a subgraph B with one node per node of G. Computation in B proceeds in exactly the reverse of the order of computation in G, and each node of B computes (n) the derivative ∂u associated with the forward graph node u (i). This is done ∂u(i) using the chain rule with respect to scalar output u(n) : ∂u(n) = ∂u(j)

i:j ∈P a(u(i) )

∂u (n) ∂u(i) ∂u (i) ∂u(j)

(6.49)

as speciﬁed by Algorithm 6.2. The subgraph B contains exactly one edge for each edge from node u(j) to node u (i) of G. The edge from u (j) to u(i) is associated with (i) the computation of ∂u . In addition, a dot product is performed for each node, ∂u(j ) between the gradient already computed with respect to nodes u(i) that are children 208

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( i)

of u (j) and the vector containing the partial derivatives ∂u for the same children ∂u(j ) ( i ) nodes u . To summarize, the amount of computation required for performing the back-propagation scales linearly with the number of edges in G, where the computation for each edge corresponds to computing a partial derivative (of one node with respect to one of its parents) as well as performing one multiplication and one addition. Below, we generalize this analysis to tensor-valued nodes, which is just a way to group multiple scalar values in the same node and enable more eﬃcient implementations. Algorithm 6.2 Simpliﬁed version of the back-propagation algorithm for computing the derivatives of u(n) with respect to the variables in the graph. This example is intended to further understanding by showing a simpliﬁed case where all variables are scalars, and we wish to compute the derivatives with respect to u(1) , . . . , u(ni ) . This simpliﬁed version computes the derivatives of all nodes in the graph. The computational cost of this algorithm is proportional to the number of edges in the graph, assuming that the partial derivative associated with each edge requires a constant time. This is of the same order as the number of computations for ( i) the forward propagation. Each ∂u is a function of the parents u (j) of u(i) , thus ( ∂u j ) linking the nodes of the forward graph to those added for the back-propagation graph. Run forward propagation (Algorithm 6.1 for this example) to obtain the activations of the network Initialize grad_table, a data structure that will store the derivatives that have been computed. The entry grad_table[u(i)] will store the computed value of ∂u(n) . ∂u(i) grad_table[∂u(n) ] ← 1 for j = n − 1 down to 1 do (n) ∂u(n) ∂u(i) = The next line computes ∂u using stored values: (i)) ( j ) i : j ∈ P a ( u ∂u ∂u(i) ∂u (j ) (i) grad_table[u(j) ] ← i:j∈P a(u(i) ) grad_table[u(i)] ∂u ∂u(j ) end for return {grad_table[u(i)] | i = 1, . . . , n i} The back-propagation algorithm is designed to reduce the number of common subexpressions without regard to memory. Speciﬁcally, it performs on the order of one Jacobian product per node in the graph. This can be seen from the fact in Algorithm 6.2 that backprop visits each edge from node u(j) to node u(i) of (j ) the graph exactly once in order to obtain the associated partial derivative ∂u . ∂u (i) Back-propagation thus avoids the exponential explosion in repeated subexpressions. 209

CHAPTER 6. DEEP FEEDFORWARD NETWORKS

However, other algorithms may be able to avoid more subexpressions by performing simpliﬁcations on the computational graph, or may be able to conserve memory by recomputing rather than storing some subexpressions. We will revisit these ideas after describing the back-propagation algorithm itself.

6.5.4

Back-Propagation Computation in Fully-Connected MLP

To clarify the above deﬁnition of the back-propagation computation, let us consider the speciﬁc graph associated with a fully-connected multi-layer MLP. Algorithm 6.3 ﬁrst shows the forward propagation, which maps parameters to y, y ) associated with a single (input,target) training example the supervised loss L(ˆ (x, y ), with yˆ the output of the neural network when x is provided in input. Algorithm 6.4 then shows the corresponding computation to be done for applying the back-propagation algorithm to this graph. Algorithm 6.3 and Algorithm 6.4 are demonstrations that are chosen to be simple and straightforward to understand. However, they are specialized to one speciﬁc problem. Modern software implementations are based on the generalized form of backpropagation described in Sec. 6.5.6 below, which can accommodate any computational graph by explicitly manipulating a data structure for representing symbolic computation.

6.5.5

Symbol-to-Symbol Derivatives

Algebraic expressions and computational graphs both operate on symbols, or variables that do not have speciﬁc values. These algebraic and graph-based representations are called symbolic representations. When we actually use or train a neural network, we must assign speciﬁc values to these symbols. We replace a symbolic input to the network x with a speciﬁc numeric value, such as [1.2, 3.765, −1.8] .

Some approaches to back-propagation take a computational graph and a set of numerical values for the inputs to the graph, then return a set of numerical values describing the gradient at those input values. We call this approach “symbolto-number” diﬀerentiation. This is the approach used by libraries such as Torch (Collobert et al., 2011b) and Caﬀe (Jia, 2013). Another approach is to take a computational graph and add additional nodes to the graph that provide a symbolic description of the desired derivatives. This 210

CHAPTER 6. DEEP FEEDFORWARD NETWORKS

z

f y

f x

f w

Figure 6.9: A computational graph that results in repeated subexpressions when computing the gradient. Let w ∈ R be the input to the graph. We use the same function f : R → R as the operation that we apply at every step of a chain: x = f(w), y = f ( x), z = f ( y). ∂z To compute ∂w , we apply Eq. 6.44 and obtain: ∂z ∂w ∂z ∂y ∂x = ∂y ∂x ∂w =f (y)f (x)f (w)

(6.50) (6.51) (6.52)

=f (f (f (w)))f (f (w))f (w)

(6.53)

Eq. 6.52 suggests an implementation in which we compute the value of f (w) only once and store it in the variable x. This is the approach taken by the back-propagation algorithm. An alternative approach is suggested by Eq. 6.53, where the subexpression f(w) appears more than once. In the alternative approach, f (w) is recomputed each time it is needed. When the memory required to store the value of these expressions is low, the back-propagation approach of Eq. 6.52 is clearly preferable because of its reduced runtime. However, Eq. 6.53 is also a valid implementation of the chain rule, and is useful when memory is limited.

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Algorithm 6.3 Forward propagation through a typical deep neural network and y, y ) depends on the output yˆ the computation of the cost function. The loss L(ˆ and on the target y (see Sec. 6.2.1.1 for examples of loss functions). To obtain the total cost J , the loss may be added to a regularizer Ω(θ), where θ contains all the parameters (weights and biases). Algorithm 6.4 shows how to compute gradients of J with respect to parameters W and b . For simplicity, this demonstration uses only a single input example x. Practical applications should use a minibatch. See Sec. 6.5.7 for a more realistic demonstration. Require: Network depth, l Require: W (i), i ∈ {1, . . . , l}, the weight matrices of the model Require: b(i) , i ∈ {1, . . . , l}, the bias parameters of the model Require: x, the input to process Require: y, the target output h(0) = x for k = 1, . . . , l do a (k) = b(k) + W (k) h(k−1) h(k) = f (a (k)) end for y ˆ = h(l) J = L(ˆ y, y) + λΩ(θ )

is the approach taken by Theano (Bergstra et al., 2010; Bastien et al., 2012) and TensorFlow (Abadi et al., 2015). An example of how this approach works is illustrated in Fig. 6.10. The primary advantage of this approach is that the derivatives are described in the same language as the original expression. Because the derivatives are just another computational graph, it is possible to run back-propagation again, diﬀerentiating the derivatives in order to obtain higher derivatives. Computation of higher-order derivatives is described in Sec. 6.5.10. We will use the latter approach and describe the back-propagation algorithm in terms of constructing a computational graph for the derivatives. Any subset of the graph may then be evaluated using speciﬁc numerical values at a later time. This allows us to avoid specifying exactly when each operation should be computed. Instead, a generic graph evaluation engine can evaluate every node as soon as its parents’ values are available. The description of the symbol-to-symbol based approach subsumes the symbolto-number approach. The symbol-to-number approach can be understood as performing exactly the same computations as are done in the graph built by the symbol-to-symbol approach. The key diﬀerence is that the symbol-to-number 212

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Algorithm 6.4 Backward computation for the deep neural network of Algorithm 6.3, which uses in addition to the input x a target y. This computation yields the gradients on the activations a(k) for each layer k, starting from the output layer and going backwards to the ﬁrst hidden layer. From these gradients, which can be interpreted as an indication of how each layer’s output should change to reduce error, one can obtain the gradient on the parameters of each layer. The gradients on weights and biases can be immediately used as part of a stochastic gradient update (performing the update right after the gradients have been computed) or used with other gradient-based optimization methods. After the forward computation, compute the gradient on the output layer: ˆ y) g ← ∇yˆJ = ∇yˆL(y, for k = l, l − 1, . . . , 1 do Convert the gradient on the layer’s output into a gradient into the prenonlinearity activation (element-wise multiplication if f is element-wise): g ← ∇ a(k) J = g f (a(k)) Compute gradients on weights and biases (including the regularization term, where needed): ∇b(k) J = g + λ∇ b(k) Ω(θ) ∇W (k)J = g h(k−1) + λ∇W (k) Ω(θ) Propagate the gradients w.r.t. the next lower-level hidden layer’s activations: g ← ∇ h(k−1) J = W (k) g end for

213

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z

f

z

f

f y

f

y

dz dy

f

f x

f

x

dy dx

×

dz dx

f

f w

w

dx dw

×

dz dw

Figure 6.10: An example of the symbol-to-symbol approach to computing derivatives. In this approach, the back-propagation algorithm does not need to ever access any actual speciﬁc numeric values. Instead, it adds nodes to a computational graph describing how to compute these derivatives. A generic graph evaluation engine can later compute the derivatives for any speciﬁc numeric values. (Left) In this example, we begin with a graph representing z = f(f (f(w))). (Right) We run the back-propagation algorithm, instructing dz it to construct the graph for the expression corresponding to dw . In this example, we do not explain how the back-propagation algorithm works. The purpose is only to illustrate what the desired result is: a computational graph with a symbolic description of the derivative.

approach does not expose the graph.

6.5.6

General Back-Propagation

The back-propagation algorithm is very simple. To compute the gradient of some scalar z with respect to one of its ancestors x in the graph, we begin by observing that the gradient with respect to z is given by dz dz = 1. We can then compute the gradient with respect to each parent of z in the graph by multiplying the current gradient by the Jacobian of the operation that produced z. We continue multiplying by Jacobians traveling backwards through the graph in this way until we reach x. For any node that may be reached by going backwards from z through two or more paths, we simply sum the gradients arriving from diﬀerent paths at that node. More formally, each node in the graph G corresponds to a variable. To achieve maximum generality, we describe this variable as being a tensor V. Tensor can 214

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in general have any number of dimensions, and subsume scalars, vectors, and matrices. We assume that each variable V is associated with the following subroutines: • get_operation(V): This returns the operation that computes V, represented by the edges coming into V in the computational graph. For example, there may be a Python or C++ class representing the matrix multiplication operation, and the get_operation function. Suppose we have a variable that is created by matrix multiplication, C = AB. Then get_operation (V) returns a pointer to an instance of the corresponding C++ class. • get_consumers(V, G): This returns the list of variables that are children of V in the computational graph G . • get_inputs(V, G ): This returns the list of variables that are parents of V in the computational graph G . Each operation op is also associated with a bprop operation. This bprop operation can compute a Jacobian-vector product as described by Eq. 6.47. This is how the back-propagation algorithm is able to achieve great generality. Each operation is responsible for knowing how to back-propagate through the edges in the graph that it participates in. For example, we might use a matrix multiplication operation to create a variable C = AB. Suppose that the gradient of a scalar z with respect to C is given by G. The matrix multiplication operation is responsible for deﬁning two back-propagation rules, one for each of its input arguments. If we call the bprop method to request the gradient with respect to A given that the gradient on the output is G , then the bprop method of the matrix multiplication operation must state that the gradient with respect to A is given by GB . Likewise, if we call the bprop method to request the gradient with respect to B, then the matrix operation is responsible for implementing the bprop method and specifying that the desired gradient is given by AG. The back-propagation algorithm itself does not need to know any diﬀerentiation rules. It only needs to call each operation’s bprop rules with the right arguments. Formally, op.bprop (inputs, X, G) must return (6.54) (∇X op.f(inputs)i ) G i, i

which is just an implementation of the chain rule as expressed in Eq. 6.47. Here, inputs is a list of inputs that are supplied to the operation, op.f is the mathematical function that the operation implements, X is the input whose gradient we wish to compute, and G is the gradient on the output of the operation. 215

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The op.bprop method should always pretend that all of its inputs are distinct from each other, even if they are not. For example, if the mul operator is passed two copies of x to compute x2, the op.bprop method should still return x as the derivative with respect to both inputs. The back-propagation algorithm will later add both of these arguments together to obtain 2x, which is the correct total derivative on x. Software implementations of back-propagation usually provide both the operations and their bprop methods, so that users of deep learning software libraries are able to back-propagate through graphs built using common operations like matrix multiplication, exponents, logarithms, and so on. Software engineers who build a new implementation of back-propagation or advanced users who need to add their own operation to an existing library must usually derive the op.bprop method for any new operations manually. The back-propagation algorithm is formally described in Algorithm 6.5. Algorithm 6.5 The outermost skeleton of the back-propagation algorithm. This portion does simple setup and cleanup work. Most of the important work happens in the build_grad subroutine of Algorithm 6.6 . Require: T, the target set of variables whose gradients must be computed. Require: G, the computational graph Require: z, the variable to be diﬀerentiated Let G be G pruned to contain only nodes that are ancestors of z and descendents of nodes in T. Initialize grad_table, a data structure associating tensors to their gradients grad_table[z] ← 1 for V in T do build_grad(V, G , G , grad_table) end for Return grad_table restricted to T In Sec. 6.5.2, we motivated back-propagation as a strategy for avoiding computing the same subexpression in the chain rule multiple times. The naive algorithm could have exponential runtime due to these repeated subexpressions. Now that we have speciﬁed the back-propagation algorithm, we can understand its computational cost. If we assume that each operation evaluation has roughly the same cost, then we may analyze the computational cost in terms of the number of operations executed. Keep in mind here that we refer to an operation as the fundamental unit of our computational graph, which might actually consist of very 216

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Algorithm 6.6 The inner loop subroutine build_grad ( V, G , G , grad_table) of the back-propagation algorithm, called by the back-propagation algorithm deﬁned in Algorithm 6.5. Require: V, the variable whose gradient should be added to G and grad_table. Require: G, the graph to modify. Require: G , the restriction of G to nodes that participate in the gradient. Require: grad_table, a data structure mapping nodes to their gradients if V is in grad_table then Return grad_table[V] end if i←1 for C in get_consumers(V, G ) do op ← get_operation(C) D ← build_grad(C, G , G , grad_table) G(i) ← op.bprop(get_inputs(C, G ), V, D) i←i+1 end for G ← i G(i) grad_table[V] = G Insert G and the operations creating it into G Return G many arithmetic operations (for example, we might have a graph that treats matrix multiplication as a single operation). Computing a gradient in a graph with n nodes will never execute more than O(n2 ) operations or store the output of more than O(n2) operations. Here we are counting operations in the computational graph, not individual operations executed by the underlying hardware, so it is important to remember that the runtime of each operation may be highly variable. For example, multiplying two matrices that each contain millions of entries might correspond to a single operation in the graph. We can see that computing the gradient requires as most O(n 2) operations because the forward propagation stage will at worst execute all n nodes in the original graph (depending on which values we want to compute, we may not need to execute the entire graph). The back-propagation algorithm adds one Jacobian-vector product, which should be expressed with O(1) nodes, per edge in the original graph. Because the computational graph is a directed acyclic graph it has at most O(n2) edges. For the kinds of graphs that are commonly used in practice, the situation is even better. Most neural network cost functions are roughly chain-structured, causing back-propagation to have O(n) cost. This is far 217

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better than the naive approach, which might need to execute exponentially many nodes. This potentially exponential cost can be seen by expanding and rewriting the recursive chain rule (Eq. 6.49) non-recursively: ∂u (n) = ∂u(j)

path (u (π1 ),u (π2 ) ,...,u(π t )), from π1=j to π t=n

t ∂u(πk ) . (πk−1) ∂u k=2

(6.55)

Since the number of paths from node j to node n can grow up to exponentially in the length of these paths, the number of terms in the above sum, which is the number of such paths, can grow exponentially with the depth of the forward propagation graph. This large cost would be incurred because the same computation for ∂u(i) would be redone many times. To avoid such recomputation, we can think ∂u(j ) of back-propagation as a table-ﬁlling algorithm that takes advantage of storing (n) intermediate results ∂u . Each node in the graph has a corresponding slot in a ∂u (i) table to store the gradient for that node. By ﬁlling in these table entries in order, back-propagation avoids repeating many common subexpressions. This table-ﬁlling strategy is sometimes called dynamic programming.

6.5.7

Example: Back-Propagation for MLP Training

As an example, we walk through the back-propagation algorithm as it is used to train a multilayer perceptron. Here we develop a very simple multilayer perception with a single hidden layer. To train this model, we will use minibatch stochastic gradient descent. The back-propagation algorithm is used to compute the gradient of the cost on a single minibatch. Speciﬁcally, we use a minibatch of examples from the training set formatted as a design matrix X and a vector of associated class labels y. The network computes a layer of hidden features H = max{0, XW (1) }. To simplify the presentation we do not use biases in this model. We assume that our graph language includes a relu operation that can compute max{0, Z} elementwise. The predictions of the unnormalized log probabilities over classes are then given by HW (2). We assume that our graph language includes a cross_entropy operation that computes the cross-entropy between the targets y and the probability distribution deﬁned by these unnormalized log probabilities. The resulting crossentropy deﬁnes the cost JMLE . Minimizing this cross-entropy performs maximum likelihood estimation of the classiﬁer. However, to make this example more realistic,

218

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J

J MLE

cross_entropy

U (2)

+

y

u(8)

matmul ×

H

W (2)

relu

sqr

U (5)

sum

u(6)

u(7)

λ

+

U (1)

matmul

X

W (1)

sqr

U (3)

sum

u(4)

Figure 6.11: The computational graph used to compute the cost used to train our example of a single-layer MLP using the cross-entropy loss and weight decay.

we also include a regularization term. The total cost (1)2 (2)2 J = JMLE + λ Wi,j + Wi,j i,j

(6.56)

i,j

consists of the cross-entropy and a weight decay term with coeﬃcient λ. The computational graph is illustrated in Fig. 6.11. The computational graph for the gradient of this example is large enough that it would be tedious to draw or to read. This demonstrates one of the beneﬁts of the back-propagation algorithm, which is that it can automatically generate gradients that would be straightforward but tedious for a software engineer to derive manually. We can roughly trace out the behavior of the back-propagation algorithm by looking at the forward propagation graph in Fig. 6.11. To train, we wish to compute both ∇ W (1) J and ∇W (2)J . There are two diﬀerent paths leading backward from J to the weights: one through the cross-entropy cost, and one through the weight decay cost. The weight decay cost is relatively simple; it will always contribute 2λW (i) to the gradient on W (i). 219

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The other path through the cross-entropy cost is slightly more complicated. Let G be the gradient on the unnormalized log probabilities U (2) provided by the cross_entropy operation. The back-propagation algorithm now needs to explore two diﬀerent branches. On the shorter branch, it adds H G to the gradient on W (2), using the back-propagation rule for the second argument to the matrix multiplication operation. The other branch corresponds to the longer chain descending further along the network. First, the back-propagation algorithm computes ∇H J = GW (2) using the back-propagation rule for the ﬁrst argument to the matrix multiplication operation. Next, the relu operation uses its backpropagation rule to zero out components of the gradient corresponding to entries of U (1) that were less than 0. Let the result be called G. The last step of the back-propagation algorithm is to use the back-propagation rule for the second argument of the matmul operation to add X G to the gradient on W (1) . After these gradients have been computed, it is the responsibility of the gradient descent algorithm, or another optimization algorithm, to use these gradients to update the parameters. For the MLP, the computational cost is dominated by the cost of matrix multiplication. During the forward propagation stage, we multiply by each weight matrix, resulting in O(w) multiply-adds, where w is the number of weights. During the backward propagation stage, we multiply by the transpose of each weight matrix, which has the same computational cost. The main memory cost of the algorithm is that we need to store the input to the nonlinearity of the hidden layer. This value is stored from the time it is computed until the backward pass has returned to the same point. The memory cost is thus O(mnh ), where m is the number of examples in the minibatch and nh is the number of hidden units.

6.5.8

Complications

Our description of the back-propagation algorithm here is simpler than the implementations actually used in practice. As noted above, we have restricted the deﬁnition of an operation to be a function that returns a single tensor. Most software implementations need to support operations that can return more than one tensor. For example, if we wish to compute both the maximum value in a tensor and the index of that value, it is best to compute both in a single pass through memory, so it is most eﬃcient to implement this procedure as a single operation with two outputs. We have not described how to control the memory consumption of backpropagation. Back-propagation often involves summation of many tensors together. 220

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In the naive approach, each of these tensors would be computed separately, then all of them would be added in a second step. The naive approach has an overly high memory bottleneck that can be avoided by maintaining a single buﬀer and adding each value to that buﬀer as it is computed. Real-world implementations of back-propagation also need to handle various data types, such as 32-bit ﬂoating point, 64-bit ﬂoating point, and integer values. The policy for handling each of these types takes special care to design. Some operations have undeﬁned gradients, and it is important to track these cases and determine whether the gradient requested by the user is undeﬁned. Various other technicalities make real-world diﬀerentiation more complicated. These technicalities are not insurmountable, and this chapter has described the key intellectual tools needed to compute derivatives, but it is important to be aware that many more subtleties exist.

6.5.9

Diﬀerentiation outside the Deep Learning Community

The deep learning community has been somewhat isolated from the broader computer science community and has largely developed its own cultural attitudes concerning how to perform diﬀerentiation. More generally, the ﬁeld of automatic diﬀerentiation is concerned with how to compute derivatives algorithmically. The back-propagation algorithm described here is only one approach to automatic diﬀerentiation. It is a special case of a broader class of techniques called reverse mode accumulation. Other approaches evaluate the subexpressions of the chain rule in diﬀerent orders. In general, determining the order of evaluation that results in the lowest computational cost is a diﬃcult problem. Finding the optimal sequence of operations to compute the gradient is NP-complete (Naumann, 2008), in the sense that it may require simplifying algebraic expressions into their least expensive form. For example, suppose we have variables p1 , p2, . . . , pn representing probabilities and variables z1 , z2 , . . . , zn representing unnormalized log probabilities. Suppose we deﬁne exp(zi) qi = , (6.57) i exp(z i)

where we build the softmax function out of exponentiation, summation and division operations, and construct a cross-entropy loss J = − i pi log qi. A human mathematician can observe that the derivative of J with respect to z i takes a very simple form: qi − pi. The back-propagation algorithm is not capable of simplifying the gradient this way, and will instead explicitly propagate gradients through all of 221

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the logarithm and exponentiation operations in the original graph. Some software libraries such as Theano (Bergstra et al., 2010; Bastien et al., 2012) are able to perform some kinds of algebraic substitution to improve over the graph proposed by the pure back-propagation algorithm. When the forward graph G has a single output node and each partial derivative can be computed with a constant amount of computation, back-propagation guarantees that the number of computations for the gradient computation is of the same order as the number of computations for the forward computation: this (i) can be seen in Algorithm 6.2 because each local partial derivative ∂u needs ∂u(j ) to be computed only once along with an associated multiplication and addition for the recursive chain-rule formulation (Eq. 6.49). The overall computation is therefore O(# edges). However, it can potentially be reduced by simplifying the computational graph constructed by back-propagation, and this is an NP-complete task. Implementations such as Theano and TensorFlow use heuristics based on matching known simpliﬁcation patterns in order to iteratively attempt to simplify the graph. We deﬁned back-propagation only for the computation of a gradient of a scalar output but back-propagation can be extended to compute a Jacobian (either of k diﬀerent scalar nodes in the graph, or of a tensor-valued node containing k values). A naive implementation may then need k times more computation: for each scalar internal node in the original forward graph, the naive implementation computes k gradients instead of a single gradient. When the number of outputs of the graph is larger than the number of inputs, it is sometimes preferable to use another form of automatic diﬀerentiation called forward mode accumulation. Forward mode computation has been proposed for obtaining real-time computation of gradients in recurrent networks, for example (Williams and Zipser, 1989). This also avoids the need to store the values and gradients for the whole graph, trading oﬀ computational eﬃciency for memory. The relationship between forward mode and backward mode is analogous to the relationship between left-multiplying versus right-multiplying a sequence of matrices, such as ∂u(i) ∂u(j )

ABCD,

(6.58)

where the matrices can be thought of as Jacobian matrices. For example, if D is a column vector while A has many rows, this corresponds to a graph with a single output and many inputs, and starting the multiplications from the end and going backwards only requires matrix-vector products. This corresponds to the backward mode. Instead, starting to multiply from the left would involve a series of matrix-matrix products, which makes the whole computation much more expensive. However, if A has fewer rows than D has columns, it is cheaper to run the multiplications left-to-right, corresponding to the forward mode. 222

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In many communities outside of machine learning, it is more common to implement diﬀerentiation software that acts directly on traditional programming language code, such as Python or C code, and automatically generates programs that diﬀerent functions written in these languages. In the deep learning community, computational graphs are usually represented by explicit data structures created by specialized libraries. The specialized approach has the drawback of requiring the library developer to deﬁne the bprop methods for every operation and limiting the user of the library to only those operations that have been deﬁned. However, the specialized approach also has the beneﬁt of allowing customized back-propagation rules to be developed for each operation, allowing the developer to improve speed or stability in non-obvious ways that an automatic procedure would presumably be unable to replicate. Back-propagation is therefore not the only way or the optimal way of computing the gradient, but it is a very practical method that continues to serve the deep learning community very well. In the future, diﬀerentiation technology for deep networks may improve as deep learning practitioners become more aware of advances in the broader ﬁeld of automatic diﬀerentiation.

6.5.10

Higher-Order Derivatives

Some software frameworks support the use of higher-order derivatives. Among the deep learning software frameworks, this includes at least Theano and TensorFlow. These libraries use the same kind of data structure to describe the expressions for derivatives as they use to describe the original function being diﬀerentiated. This means that the symbolic diﬀerentiation machinery can be applied to derivatives. In the context of deep learning, it is rare to compute a single second derivative of a scalar function. Instead, we are usually interested in properties of the Hessian matrix. If we have a function f : R n → R, then the Hessian matrix is of size n × n. In typical deep learning applications, n will be the number of parameters in the model, which could easily number in the billions. The entire Hessian matrix is thus infeasible to even represent. Instead of explicitly computing the Hessian, the typical deep learning approach is to use Krylov methods. Krylov methods are a set of iterative techniques for performing various operations like approximately inverting a matrix or ﬁnding approximations to its eigenvectors or eigenvalues, without using any operation other than matrix-vector products. In order to use Krylov methods on the Hessian, we only need to be able to compute the product between the Hessian matrix H and an arbitrary vector v . A 223

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straightforward technique (Christianson, 1992) for doing so is to compute Hv = ∇ x (∇x f (x)) v . (6.59)

Both of the gradient computations in this expression may be computed automatically by the appropriate software library. Note that the outer gradient expression takes the gradient of a function of the inner gradient expression. If v is itself a vector produced by a computational graph, it is important to specify that the automatic diﬀerentiation software should not diﬀerentiate through the graph that produced v . While computing the Hessian is usually not advisable, it is possible to do with Hessian vector products. One simply computes He(i) for all i = 1, . . . , n, where (i) e(i) is the one-hot vector with ei = 1 and all other entries equal to 0.

6.6

Historical Notes

Feedforward networks can be seen as eﬃcient nonlinear function approximators based on using gradient descent to minimize the error in a function approximation. From this point of view, the modern feedforward network is the culmination of centuries of progress on the general function approximation task. The chain rule that underlies the back-propagation algorithm was invented in the 17th century (Leibniz, 1676; L’Hôpital, 1696). Calculus and algebra have long been used to solve optimization problems in closed form, but gradient descent was not introduced as a technique for iteratively approximating the solution to optimization problems until the 19th century (Cauchy, 1847). Beginning in the 1940s, these function approximation techniques were used to motivate machine learning models such as the perceptron. However, the earliest models were based on linear models. Critics including Marvin Minsky pointed out several of the ﬂaws of the linear model family, such as it inability to learn the XOR function, which led to a backlash against the entire neural network approach. Learning nonlinear functions required the development of a multilayer perceptron and a means of computing the gradient through such a model. Eﬃcient applications of the chain rule based on dynamic programming began to appear in the 1960s and 1970s, mostly for control applications (Kelley, 1960; Bryson and Denham, 1961; Dreyfus, 1962; Bryson and Ho, 1969; Dreyfus, 1973) but also for sensitivity analysis (Linnainmaa, 1976). Werbos (1981) proposed applying these techniques to training artiﬁcial neural networks. The idea was ﬁnally developed in practice after being independently rediscovered in diﬀerent ways (LeCun, 1985; Parker, 224

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1985; Rumelhart et al., 1986a). The book Parallel Distributed Processing presented the results of some of the ﬁrst successful experiments with back-propagation in a chapter (Rumelhart et al., 1986b) that contributed greatly to the popularization of back-propagation and initiated a very active period of research in multi-layer neural networks. However, the ideas put forward by the authors of that book and in particular by Rumelhart and Hinton go much beyond back-propagation. They include crucial ideas about the possible computational implementation of several central aspects of cognition and learning, which came under the name of “connectionism” because of the importance given the connections between neurons as the locus of learning and memory. In particular, these ideas include the notion of distributed representation (Hinton et al., 1986). Following the success of back-propagation, neural network research gained popularity and reached a peak in the early 1990s. Afterwards, other machine learning techniques became more popular until the modern deep learning renaissance that began in 2006. The core ideas behind modern feedforward networks have not changed substantially since the 1980s. The same back-propagation algorithm and the same approaches to gradient descent are still in use. Most of the improvement in neural network performance from 1986 to 2015 can be attributed to two factors. First, larger datasets have reduced the degree to which statistical generalization is a challenge for neural networks. Second, neural networks have become much larger, due to more powerful computers, and better software infrastructure. However, a small number of algorithmic changes have improved the performance of neural networks noticeably. One of these algorithmic changes was the replacement of mean squared error with the cross-entropy family of loss functions. Mean squared error was popular in the 1980s and 1990s, but was gradually replaced by cross-entropy losses and the principle of maximum likelihood as ideas spread between the statistics community and the machine learning community. The use of cross-entropy losses greatly improved the performance of models with sigmoid and softmax outputs, which had previously suﬀered from saturation and slow learning when using the mean squared error loss. The other major algorithmic change that has greatly improved the performance of feedforward networks was the replacement of sigmoid hidden units with piecewise linear hidden units, such as rectiﬁed linear units. Rectiﬁcation using the max{0, z} function was introduced in early neural network models and dates back at least as far as the Cognitron and Neocognitron (Fukushima, 1975, 1980). These early models did not use rectiﬁed linear units, but instead applied rectiﬁcation to 225

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nonlinear functions. Despite the early popularity of rectiﬁcation, rectiﬁcation was largely replaced by sigmoids in the 1980s, perhaps because sigmoids perform better when neural networks are very small. As of the early 2000s, rectiﬁed linear units were avoided due to a somewhat superstitious belief that activation functions with non-diﬀerentiable points must be avoided. This began to change in about 2009. Jarrett et al. (2009) observed that “using a rectifying nonlinearity is the single most important factor in improving the performance of a recognition system” among several diﬀerent factors of neural network architecture design. For small datasets, Jarrett et al. (2009) observed that using rectifying nonlinearities is even more important than learning the weights of the hidden layers. Random weights are suﬃcient to propagate useful information through a rectiﬁed linear network, allowing the classiﬁer layer at the top to learn how to map diﬀerent feature vectors to class identities. When more data is available, learning begins to extract enough useful knowledge to exceed the performance of randomly chosen parameters. Glorot et al. (2011a) showed that learning is far easier in deep rectiﬁed linear networks than in deep networks that have curvature or two-sided saturation in their activation functions. Rectiﬁed linear units are also of historical interest because they show that neuroscience has continued to have an inﬂuence on the development of deep learning algorithms. Glorot et al. (2011a) motivate rectiﬁed linear units from biological considerations. The half-rectifying nonlinearity was intended to capture these properties of biological neurons: 1) For some inputs, biological neurons are completely inactive. 2) For some inputs, a biological neuron’s output is proportional to its input. 3) Most of the time, biological neurons operate in the regime where they are inactive (i.e., they should have sparse activations). When the modern resurgence of deep learning began in 2006, feedforward networks continued to have a bad reputation. From about 2006-2012, it was widely believed that feedforward networks would not perform well unless they were assisted by other models, such as probabilistic models. Today, it is now known that with the right resources and engineering practices, feedforward networks perform very well. Today, gradient-based learning in feedforward networks is used as a tool to develop probabilistic models, such as the variational autoencoder and generative adversarial networks, described in Chapter 20. Rather than being viewed as an unreliable technology that must be supported by other techniques, gradient-based learning in feedforward networks has been viewed since 2012 as a powerful technology that may be applied to many other machine learning tasks. In 2006, the community used unsupervised learning to support supervised learning, and now, ironically, it is more common to use supervised learning to support unsupervised learning. 226

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Feedforward networks continue to have unfulﬁlled potential. In the future, we expect they will be applied to many more tasks, and that advances in optimization algorithms and model design will improve their performance even further. This chapter has primarily described the neural network family of models. In the subsequent chapters, we turn to how to use these models—how to regularize and train them.

227

Chapter 7

Regularization for Deep Learning A central problem in machine learning is how to make an algorithm that will perform well not just on the training data, but also on new inputs. Many strategies used in machine learning are explicitly designed to reduce the test error, possibly at the expense of increased training error. These strategies are known collectively as regularization. As we will see there are a great many forms of regularization available to the deep learning practitioner. In fact, developing more eﬀective regularization strategies has been one of the major research eﬀorts in the ﬁeld. Chapter 5 introduced the basic concepts of generalization, underﬁtting, overﬁtting, bias, variance and regularization. If you are not already familiar with these notions, please refer to that chapter before continuing with this one. In this chapter, we describe regularization in more detail, focusing on regularization strategies for deep models or models that may be used as building blocks to form deep models. Some sections of this chapter deal with standard concepts in machine learning. If you are already familiar with these concepts, feel free to skip the relevant sections. However, most of this chapter is concerned with the extension of these basic concepts to the particular case of neural networks. In Sec. 5.2.2, we deﬁned regularization as “any modiﬁcation we make to a learning algorithm that is intended to reduce its generalization error but not its training error.” There are many regularization strategies. Some put extra constraints on a machine learning model, such as adding restrictions on the parameter values. Some add extra terms in the objective function that can be thought of as corresponding to a soft constraint on the parameter values. If chosen carefully, these extra constraints and penalties can lead to improved performance 228

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on the test set. Sometimes these constraints and penalties are designed to encode speciﬁc kinds of prior knowledge. Other times, these constraints and penalties are designed to express a generic preference for a simpler model class in order to promote generalization. Sometimes penalties and constraints are necessary to make an underdetermined problem determined. Other forms of regularization, known as ensemble methods, combine multiple hypotheses that explain the training data. In the context of deep learning, most regularization strategies are based on regularizing estimators. Regularization of an estimator works by trading increased bias for reduced variance. An eﬀective regularizer is one that makes a proﬁtable trade, reducing variance signiﬁcantly while not overly increasing the bias. When we discussed generalization and overﬁtting in Chapter 5, we focused on three situations, where the model family being trained either (1) excluded the true data generating process—corresponding to underﬁtting and inducing bias, or (2) matched the true data generating process, or (3) included the generating process but also many other possible generating processes—the overﬁtting regime where variance rather than bias dominates the estimation error. The goal of regularization is to take a model from the third regime into the second regime. In practice, an overly complex model family does not necessarily include the target function or the true data generating process, or even a close approximation of either. We almost never have access to the true data generating process so we can never know for sure if the model family being estimated includes the generating process or not. However, most applications of deep learning algorithms are to domains where the true data generating process is almost certainly outside the model family. Deep learning algorithms are typically applied to extremely complicated domains such as images, audio sequences and text, for which the true generation process essentially involves simulating the entire universe. To some extent, we are always trying to ﬁt a square peg (the data generating process) into a round hole (our model family). What this means is that controlling the complexity of the model is not a simple matter of ﬁnding the model of the right size, with the right number of parameters. Instead, we might ﬁnd—and indeed in practical deep learning scenarios, we almost always do ﬁnd—that the best ﬁtting model (in the sense of minimizing generalization error) is a large model that has been regularized appropriately. We now review several strategies for how to create such a large, deep, regularized model.

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7.1

Parameter Norm Penalties

Regularization has been used for decades prior to the advent of deep learning. Linear models such as linear regression and logistic regression allow simple, straightforward, and eﬀective regularization strategies. Many regularization approaches are based on limiting the capacity of models, such as neural networks, linear regression, or logistic regression, by adding a parameter norm penalty Ω(θ) to the objective function J. We denote the regularized ˜ objective function by J: J˜(θ; X , y) = J (θ; X , y) + αΩ(θ)

(7.1)

where α ∈ [0, ∞) is a hyperparameter that weights the relative contribution of the norm penalty term, Ω, relative to the standard objective function J (x; θ). Setting α to 0 results in no regularization. Larger values of α correspond to more regularization. When our training algorithm minimizes the regularized objective function J˜ it will decrease both the original objective J on the training data and some measure of the size of the parameters θ (or some subset of the parameters). Diﬀerent choices for the parameter norm Ω can result in diﬀerent solutions being preferred. In this section, we discuss the eﬀects of the various norms when used as penalties on the model parameters. Before delving into the regularization behavior of diﬀerent norms, we note that for neural networks, we typically choose to use a parameter norm penalty Ω that penalizes only the weights of the aﬃne transformation at each layer and leaves the biases unregularized. The biases typically require less data to ﬁt accurately than the weights. Each weight speciﬁes how two variables interact. Fitting the weight well requires observing both variables in a variety of conditions. Each bias controls only a single variable. This means that we do not induce too much variance by leaving the biases unregularized. Also, regularizing the bias parameters can introduce a signiﬁcant amount of underﬁtting. We therefore use the vector w to indicate all of the weights that should be aﬀected by a norm penalty, while the vector θ denotes all of the parameters, including both w and the unregularized parameters. In the context of neural networks, it is sometimes desirable to use a separate penalty with a diﬀerent α coeﬃcient for each layer of the network. Because it can be expensive to search for the correct value of multiple hyperparameters, it is still reasonable to use the same weight decay at all layers just to reduce the size of search space. 230

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7.1.1

L2 Parameter Regularization

We have already seen, in Sec. 5.2.2, one of the simplest and most common kinds of parameter norm penalty: the L2 parameter norm penalty commonly known as weight decay. This regularization strategy drives the weights closer to the origin 1 by adding a regularization term Ω(θ) = 12 w22 to the objective function. In other academic communities, L2 regularization is also known as ridge regression or Tikhonov regularization. We can gain some insight into the behavior of weight decay regularization by studying the gradient of the regularized objective function. To simplify the presentation, we assume no bias parameter, so θ is just w. Such a model has the following total objective function: α J˜(w; X , y) = w w + J (w; X , y ), 2

(7.2)

with the corresponding parameter gradient ∇w J˜(w; X , y) = αw + ∇ w J (w; X , y).

(7.3)

To take a single gradient step to update the weights, we perform this update: w ← w − (αw + ∇w J (w; X , y)) .

(7.4)

Written another way, the update is: w ← (1 − α)w − ∇ w J (w; X , y).

(7.5)

We can see that the addition of the weight decay term has modiﬁed the learning rule to multiplicatively shrink the weight vector by a constant factor on each step, just before performing the usual gradient update. This describes what happens in a single step. But what happens over the entire course of training? We will further simplify the analysis by making a quadratic approximation to the objective function in the neighborhood of the value of the weights that obtains minimal unregularized training cost, w∗ = arg minw J(w). If the objective function is truly quadratic, as in the case of ﬁtting a linear regression model with 1

More generally, we could regularize the parameters to be near any speciﬁc point in space and, surprisingly, still get a regularization eﬀect, but better results will be obtained for a value closer to the true one, with zero being a default value that makes sense when we do not know if the correct value should be positive or negative. Since it is far more common to regularize the model parameters towards zero, we will focus on this special case in our exposition. 231

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mean squared error, then the approximation is perfect. The approximation Jˆ is given by 1 Jˆ(θ) = J (w∗) + (w − w ∗) H (w − w ∗ ), (7.6) 2 where H is the Hessian matrix of J with respect to w evaluated at w ∗. There is no ﬁrst-order term in this quadratic approximation, because w∗ is deﬁned to be a minimum, where the gradient vanishes. Likewise, because w∗ is the location of a minimum of J , we can conclude that H is positive semideﬁnite. The minimum of Jˆ occurs where its gradient ˆ w) = H (w − w ∗) ∇ w J(

(7.7)

is equal to 0. To study the eﬀect of weight decay, we modify Eq. 7.7 by adding the weight ˆ decay gradient. We can now solve for the minimum of the regularized version of J. ˜ to represent the location of the minimum. We use the variable w αw ˜ + H (w ˜ − w∗ ) = 0

(7.8)

(H + αI ) w ˜ = Hw ∗

(7.9)

w ˜ = (H + αI )−1 Hw ∗.

(7.10)

As α approaches 0, the regularized solution w˜ approaches w ∗. But what happens as α grows? Because H is real and symmetric, we can decompose it into a diagonal matrix Λ and an orthonormal basis of eigenvectors, Q, such that H = QΛQ. Applying the decomposition to Eq.7.10, we obtain: w ˜ = (QΛQ + αI )−1 QΛQ w ∗ −1 = Q(Λ + αI )Q QΛQ w∗ = Q(Λ + αI )−1 ΛQw∗ .

(7.11) (7.12) (7.13)

We see that the eﬀect of weight decay is to rescale w ∗ along the axes deﬁned by the eigenvectors of H. Speciﬁcally, the component of w ∗ that is aligned with the i i-th eigenvector of H is rescaled by a factor of λiλ+α . (You may wish to review how this kind of scaling works, ﬁrst explained in Fig. 2.3). Along the directions where the eigenvalues of H are relatively large, for example, where λ i α, the eﬀect of regularization is relatively small. However, components with λ i α will be shrunk to have nearly zero magnitude. This eﬀect is illustrated in Fig. 7.1. 232

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w2

w∗ w ˜

w1

Figure 7.1: An illustration of the eﬀect of L 2 (or weight decay) regularization on the value of the optimal w. The solid ellipses represent contours of equal value of the unregularized objective. The dotted circles represent contours of equal value of the L2 regularizer. At ˜ these competing objectives reach an equilibrium. In the ﬁrst dimension, the the point w, eigenvalue of the Hessian of J is small. The objective function does not increase much when moving horizontally away from w ∗ . Because the objective function does not express a strong preference along this direction, the regularizer has a strong eﬀect on this axis. The regularizer pulls w 1 close to zero. In the second dimension, the objective function is very sensitive to movements away from w∗ . The corresponding eigenvalue is large, indicating high curvature. As a result, weight decay aﬀects the position of w2 relatively little.

Only directions along which the parameters contribute signiﬁcantly to reducing the objective function are preserved relatively intact. In directions that do not contribute to reducing the objective function, a small eigenvalue of the Hessian tells us that movement in this direction will not signiﬁcantly increase the gradient. Components of the weight vector corresponding to such unimportant directions are decayed away through the use of the regularization throughout training. So far we have discussed weight decay in terms of its eﬀect on the optimization of an abstract, general, quadratic cost function. How do these eﬀects relate to machine learning in particular? We can ﬁnd out by studying linear regression, a model for which the true cost function is quadratic and therefore amenable to the same kind of analysis we have used so far. Applying the analysis again, we will be able to obtain a special case of the same results, but with the solution now phrased in terms of the training data. For linear regression, the cost function is

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the sum of squared errors: (Xw − y )(Xw − y ).

(7.14)

When we add L2 regularization, the objective function changes to 1 (Xw − y ) (Xw − y ) + αw w. 2

(7.15)

This changes the normal equations for the solution from w = (X X) −1X y

(7.16)

w = (X X + αI )−1 X y.

(7.17)

to 1 X X. The matrix X X in Eq. 7.16 is proportional to the covariance matrix m −1 Using L 2 regularization replaces this matrix with X X + αI in Eq. 7.17. The new matrix is the same as the original one, but with the addition of α to the diagonal. The diagonal entries of this matrix correspond to the variance of each input feature. We can see that L2 regularization causes the learning algorithm to “perceive” the input X as having higher variance, which makes it shrink the weights on features whose covariance with the output target is low compared to this added variance.

7.1.2

L1 Regularization

While L 2 weight decay is the most common form of weight decay, there are other ways to penalize the size of the model parameters. Another option is to use L1 regularization. Formally, L1 regularization on the model parameter w is deﬁned as: Ω(θ) = ||w||1 = |w i |,

(7.18)

i

that is, as the sum of absolute values of the individual parameters.2 We will now discuss the eﬀect of L1 regularization on the simple linear regression model, with no bias parameter, that we studied in our analysis of L2 regularization. In particular, we are interested in delineating the diﬀerences between L1 and L2 forms As with L 2 regularization, we could regularize the parameters towards a value that is not (o) zero, but instead towards some parameter value that case the L1 regularization would w . (In o) (o) introduce the term Ω(θ ) = ||w − w ||1 = i |w i − w i |. 2

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of regularization. As with L 2 weight decay, L 1 weight decay controls the strength of the regularization by scaling the penalty Ω using a positive hyperparameter α. Thus, the regularized objective function J˜(w; X , y) is given by J˜(w; X , y) = α||w||1 + J (w; X , y),

(7.19)

with the corresponding gradient (actually, sub-gradient): ∇w J˜(w; X , y) = αsign(w) + ∇w J (X , y; w)

(7.20)

where sign(w) is simply the sign of w applied element-wise. By inspecting Eq. 7.20, we can see immediately that the eﬀect of L1 regularization is quite diﬀerent from that of L2 regularization. Speciﬁcally, we can see that the regularization contribution to the gradient no longer scales linearly with each wi; instead it is a constant factor with a sign equal to sign(wi). One consequence of this form of the gradient is that we will not necessarily see clean algebraic solutions to quadratic approximations of J(X , y; w) as we did for L2 regularization. Our simple linear model has a quadratic cost function that we can represent via its Taylor series. Alternately, we could imagine that this is a truncated Taylor series approximating the cost function of a more sophisticated model. The gradient in this setting is given by ˆ w) = H (w − w ∗), ∇w J(

(7.21)

where, again, H is the Hessian matrix of J with respect to w evaluated at w ∗ . Because the L1 penalty does not admit clean algebraic expressions in the case of a fully general Hessian, we will also make the further simplifying assumption that the Hessian is diagonal, H = diag([H1,1 , . . . , Hn,n ]), where each Hi,i > 0. This assumption holds if the data for the linear regression problem has been preprocessed to remove all correlation between the input features, which may be accomplished using PCA. Our quadratic approximation of the L1 regularized objective function decomposes into a sum over the parameters: 1 ∗ ∗ 2 Jˆ(w; X , y) = J (w ; X , y) + H i,i(wi − w i ) + α|wi| . (7.22) 2 i

The problem of minimizing this approximate cost function has an analytical solution (for each dimension i), with the following form: α ∗ ∗ w i = sign(wi ) max |w i | − ,0 . (7.23) Hi,i 235

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Consider the situation where w∗i > 0 for all i. There are two possible outcomes: 1. The case where w ∗i ≤ Hαi,i . Here the optimal value of wi under the regularized objective is simply w i = 0. This occurs because the contribution of J (w; X , y) to the regularized objective J˜ (w; X , y) is overwhelmed—in direction i—by the L 1 regularization which pushes the value of wi to zero. 2. The case where w ∗i > Hαi,i . In this case, the regularization does not move the optimal value of wi to zero but instead it just shifts it in that direction by a distance equal to Hα . i,i

A similar process happens when w ∗i < 0, but with the L1 penalty making wi less negative by Hαi,i , or 0. In comparison to L 2 regularization, L1 regularization results in a solution that is more sparse. Sparsity in this context refers to the fact that some parameters have an optimal value of zero. The sparsity of L1 regularization is a qualitatively diﬀerent behavior than arises with L2 regularization. Eq. 7.13 gave the solution w ˜ for L2 regularization. If we revisit that equation using the assumption of a diagonal and positive deﬁnite Hessian H that we introduced for our analysis of L1 H ˜i = H i,ii,i+α w∗i . If w∗i was nonzero, then w ˜ i remains regularization, we ﬁnd that w nonzero. This demonstrates that L2 regularization does not cause the parameters to become sparse, while L1 regularization may do so for large enough α.

The sparsity property induced by L 1 regularization has been used extensively as a feature selection mechanism. Feature selection simpliﬁes a machine learning problem by choosing which subset of the available features should be used. In particular, the well known LASSO (Tibshirani, 1995) (least absolute shrinkage and selection operator) model integrates an L1 penalty with a linear model and a least squares cost function. The L1 penalty causes a subset of the weights to become zero, suggesting that the corresponding features may safely be discarded. In Sec. 5.6.1, we saw that many regularization strategies can be interpreted as MAP Bayesian inference, and that in particular, L2 regularization is equivalent to MAP Bayesian inference with a Gaussian prior on the weights. For L1 regu larization, the penalty αΩ(w) = α i |wi | used to regularize a cost function is equivalent to the log-prior term that is maximized by MAP Bayesian inference when the prior is an isotropic Laplace distribution (Eq. 3.26) over w ∈ R n : log p(w) =

i

log Laplace(w i; 0,

1 ) = −α||w||1 + n log α − n log 2. α 236

(7.24)

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From the point of view of learning via maximization with respect to w, we can ignore the log α − log 2 terms because they do not depend on w.

7.2

Norm Penalties as Constrained Optimization

Consider the cost function regularized by a parameter norm penalty: J˜(θ; X , y) = J (θ; X , y) + αΩ(θ).

(7.25)

Recall from Sec. 4.4 that we can minimize a function subject to constraints by constructing a generalized Lagrange function, consisting of the original objective function plus a set of penalties. Each penalty is a product between a coeﬃcient, called a Karush–Kuhn–Tucker (KKT) multiplier, and a function representing whether the constraint is satisﬁed. If we wanted to constrain Ω(θ) to be less than some constant k, we could construct a generalized Lagrange function L(θ, α; X , y) = J (θ; X , y) + α(Ω(θ) − k).

(7.26)

The solution to the constrained problem is given by θ∗ = arg min max L(θ, α). θ

α,α≥0

(7.27)

As described in Sec. 4.4, solving this problem requires modifying both θ and α. Sec. 4.5 provides a worked example of linear regression with an L2 constraint. Many diﬀerent procedures are possible—some may use gradient descent, while others may use analytical solutions for where the gradient is zero—but in all procedures α must increase whenever Ω(θ) > k and decrease whenever Ω( θ) < k. All positive α encourage Ω( θ) to shrink. The optimal value α ∗ will encourage Ω(θ) to shrink, but not so strongly to make Ω(θ) become less than k. To gain some insight into the eﬀect of the constraint, we can ﬁx α ∗ and view the problem as just a function of θ : θ ∗ = arg min L(θ, α∗) = arg min J (θ; X , y) + α ∗Ω(θ). θ

(7.28)

θ

˜ This is exactly the same as the regularized training problem of minimizing J. We can thus think of a parameter norm penalty as imposing a constraint on the weights. If Ω is the L2 norm, then the weights are constrained to lie in an L2 ball. If Ω is the L1 norm, then the weights are constrained to lie in a region of 237

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limited L1 norm. Usually we do not know the size of the constraint region that we impose by using weight decay with coeﬃcient α∗ because the value of α∗ does not directly tell us the value of k. In principle, one can solve for k, but the relationship between k and α∗ depends on the form of J . While we do not know the exact size of the constraint region, we can control it roughly by increasing or decreasing α in order to grow or shrink the constraint region. Larger α will result in a smaller constraint region. Smaller α will result in a larger constraint region. Sometimes we may wish to use explicit constraints rather than penalties. As described in Sec. 4.4, we can modify algorithms such as stochastic gradient descent to take a step downhill on J (θ) and then project θ back to the nearest point that satisﬁes Ω(θ ) < k. This can be useful if we have an idea of what value of k is appropriate and do not want to spend time searching for the value of α that corresponds to this k. Another reason to use explicit constraints and reprojection rather than enforcing constraints with penalties is that penalties can cause non-convex optimization procedures to get stuck in local minima corresponding to small θ. When training neural networks, this usually manifests as neural networks that train with several “dead units.” These are units that do not contribute much to the behavior of the function learned by the network because the weights going into or out of them are all very small. When training with a penalty on the norm of the weights, these conﬁgurations can be locally optimal, even if it is possible to signiﬁcantly reduce J by making the weights larger. Explicit constraints implemented by re-projection can work much better in these cases because they do not encourage the weights to approach the origin. Explicit constraints implemented by re-projection only have an eﬀect when the weights become large and attempt to leave the constraint region. Finally, explicit constraints with reprojection can be useful because they impose some stability on the optimization procedure. When using high learning rates, it is possible to encounter a positive feedback loop in which large weights induce large gradients which then induce a large update to the weights. If these updates consistently increase the size of the weights, then θ rapidly moves away from the origin until numerical overﬂow occurs. Explicit constraints with reprojection prevent this feedback loop from continuing to increase the magnitude of the weights without bound. Hinton et al. (2012c) recommend using constraints combined with a high learning rate to allow rapid exploration of parameter space while maintaining some stability. In particular, Hinton et al. (2012c) recommend a strategy introduced by Srebro and Shraibman (2005): constraining the norm of each column of the weight matrix 238

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of a neural net layer, rather than constraining the Frobenius norm of the entire weight matrix. Constraining the norm of each column separately prevents any one hidden unit from having very large weights. If we converted this constraint into a penalty in a Lagrange function, it would be similar to L2 weight decay but with a separate KKT multiplier for the weights of each hidden unit. Each of these KKT multipliers would be dynamically updated separately to make each hidden unit obey the constraint. In practice, column norm limitation is always implemented as an explicit constraint with reprojection.

7.3

Regularization and Under-Constrained Problems

In some cases, regularization is necessary for machine learning problems to be properly deﬁned. Many linear models in machine learning, including linear regression and PCA, depend on inverting the matrix XX. This is not possible whenever X X is singular. This matrix can be singular whenever the data generating distribution truly has no variance in some direction, or when no variance in observed in some direction because there are fewer examples (rows of X ) than input features (columns of X ). In this case, many forms of regularization correspond to inverting X X + αI instead. This regularized matrix is guaranteed to be invertible. These linear problems have closed form solutions when the relevant matrix is invertible. It is also possible for a problem with no closed form solution to be underdetermined. An example is logistic regression applied to a problem where the classes are linearly separable. If a weight vector w is able to achieve perfect classiﬁcation, then 2w will also achieve perfect classiﬁcation and higher likelihood. An iterative optimization procedure like stochastic gradient descent will continually increase the magnitude of w and, in theory, will never halt. In practice, a numerical implementation of gradient descent will eventually reach suﬃciently large weights to cause numerical overﬂow, at which point its behavior will depend on how the programmer has decided to handle values that are not real numbers. Most forms of regularization are able to guarantee the convergence of iterative methods applied to underdetermined problems. For example, weight decay will cause gradient descent to quit increasing the magnitude of the weights when the slope of the likelihood is equal to the weight decay coeﬃcient. The idea of using regularization to solve underdetermined problems extends beyond machine learning. The same idea is useful for several basic linear algebra problems. As we saw in Sec. 2.9, we can solve underdetermined linear equations using 239

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the Moore-Penrose pseudoinverse. Recall that one deﬁnition of the pseudoinverse X+ of a matrix X is X+ = lim (X X + αI )−1 X . α0

(7.29)

We can now recognize Eq. 7.29 as performing linear regression with weight decay. Speciﬁcally, Eq. 7.29 is the limit of Eq. 7.17 as the regularization coeﬃcient shrinks to zero. We can thus interpret the pseudoinverse as stabilizing underdetermined problems using regularization.

7.4

Dataset Augmentation

The best way to make a machine learning model generalize better is to train it on more data. Of course, in practice, the amount of data we have is limited. One way to get around this problem is to create fake data and add it to the training set. For some machine learning tasks, it is reasonably straightforward to create new fake data. This approach is easiest for classiﬁcation. A classiﬁer needs to take a complicated, high dimensional input x and summarize it with a single category identity y. This means that the main task facing a classiﬁer is to be invariant to a wide variety of transformations. We can generate new (x, y) pairs easily just by transforming the x inputs in our training set. This approach is not as readily applicable to many other tasks. For example, it is diﬃcult to generate new fake data for a density estimation task unless we have already solved the density estimation problem. Dataset augmentation has been a particularly eﬀective technique for a speciﬁc classiﬁcation problem: object recognition. Images are high dimensional and include an enormous variety of factors of variation, many of which can be easily simulated. Operations like translating the training images a few pixels in each direction can often greatly improve generalization, even if the model has already been designed to be partially translation invariant by using the convolution and pooling techniques described in Chapter 9. Many other operations such as rotating the image or scaling the image have also proven quite eﬀective. One must be careful not to apply transformations that would change the correct class. For example, optical character recognition tasks require recognizing the diﬀerence between ‘b’ and ‘d’ and the diﬀerence between ‘6’ and ‘9’, so horizontal ﬂips and 180◦ rotations are not appropriate ways of augmenting datasets for these tasks. 240

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There are also transformations that we would like our classiﬁers to be invariant to, but which are not easy to perform. For example, out-of-plane rotation can not be implemented as a simple geometric operation on the input pixels. Dataset augmentation is eﬀective for speech recognition tasks as well (Jaitly and Hinton, 2013). Injecting noise in the input to a neural network (Sietsma and Dow, 1991) can also be seen as a form of data augmentation. For many classiﬁcation and even some regression tasks, the task should still be possible to solve even if small random noise is added to the input. Neural networks prove not to be very robust to noise, however (Tang and Eliasmith, 2010). One way to improve the robustness of neural networks is simply to train them with random noise applied to their inputs. Input noise injection is part of some unsupervised learning algorithms such as the denoising autoencoder (Vincent et al., 2008). Noise injection also works when the noise is applied to the hidden units, which can be seen as doing dataset augmentation at multiple levels of abstraction. Poole et al. (2014) recently showed that this approach can be highly eﬀective provided that the magnitude of the noise is carefully tuned. Dropout, a powerful regularization strategy that will be described in Sec. 7.12, can be seen as a process of constructing new inputs by multiplying by noise. When comparing machine learning benchmark results, it is important to take the eﬀect of dataset augmentation into account. Often, hand-designed dataset augmentation schemes can dramatically reduce the generalization error of a machine learning technique. To compare the performance of one machine learning algorithm to another, it is necessary to perform controlled experiments. When comparing machine learning algorithm A and machine learning algorithm B, it is necessary to make sure that both algorithms were evaluated using the same hand-designed dataset augmentation schemes. Suppose that algorithm A performs poorly with no dataset augmentation and algorithm B performs well when combined with numerous synthetic transformations of the input. In such a case it is likely the synthetic transformations caused the improved performance, rather than the use of machine learning algorithm B. Sometimes deciding whether an experiment has been properly controlled requires subjective judgment. For example, machine learning algorithms that inject noise into the input are performing a form of dataset augmentation. Usually, operations that are generally applicable (such as adding Gaussian noise to the input) are considered part of the machine learning algorithm, while operations that are speciﬁc to one application domain (such as randomly cropping an image) are considered to be separate pre-processing steps.

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7.5

Noise Robustness

Sec. 7.4 has motivated the use of noise applied to the inputs as a dataset augmentation strategy. For some models, the addition of noise with inﬁnitesimal variance at the input of the model is equivalent to imposing a penalty on the norm of the weights (Bishop, 1995a,b). In the general case, it is important to remember that noise injection can be much more powerful than simply shrinking the parameters, especially when the noise is added to the hidden units. Noise applied to the hidden units is such an important topic as to merit its own separate discussion; the dropout algorithm described in Sec. 7.12 is the main development of that approach. Another way that noise has been used in the service of regularizing models is by adding it to the weights. This technique has been used primarily in the context of recurrent neural networks (Jim et al., 1996; Graves, 2011). This can be interpreted as a stochastic implementation of a Bayesian inference over the weights. The Bayesian treatment of learning would consider the model weights to be uncertain and representable via a probability distribution that reﬂects this uncertainty. Adding noise to the weights is a practical, stochastic way to reﬂect this uncertainty (Graves, 2011). This can also be interpreted as equivalent (under some assumptions) to a more traditional form of regularization. Adding noise to the weights has been shown to be an eﬀective regularization strategy in the context of recurrent neural networks (Jim et al., 1996; Graves, 2011). In the following, we will present an analysis of the eﬀect of weight noise on a standard feedforward neural network (as introduced in Chapter 6). We study the regression setting, where we wish to train a function yˆ(x) that maps a set of features x to a scalar using the least-squares cost function between the model predictions yˆ(x) and the true values y: J = E p(x,y) (ˆ (7.30) y(x) − y )2 .

The training set consists of m labeled examples {(x (1) , y(1) ), . . . , (x (m), y (m))}.

We now assume that with each input presentation we also include a random perturbation W ∼ N(; 0, ηI) of the network weights. Let us imagine that we have a standard l-layer MLP. We denote the perturbed model as yˆW (x). Despite the injection of noise, we are still interested in minimizing the squared error of the output of the network. The objective function thus becomes: J˜W = E p(x,y,W ) (ˆ y W (x) − y)2 (7.31) 242

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= E p(x,y,W ) yˆ2W (x) − 2yyˆW (x) + y 2 .

(7.32)

For small η, the minimization of J with added weight noise (with covariance ηI) is equivalent to minimization of J with an additional regularization term: ηEp(x,y) ∇ W yˆ(x)2 . This form of regularization encourages the parameters to go to regions of parameter space where small perturbations of the weights have a relatively small inﬂuence on the output. In other words, it pushes the model into regions where the model is relatively insensitive to small variations in the weights, ﬁnding points that are not merely minima, but minima surrounded by ﬂat regions (Hochreiter and Schmidhuber, 1995). In the simpliﬁed case of linear regression (where, for instance, yˆ(x) = w x+ b), this regularization term collapses into ηE p(x) x 2 , which is not a function of parameters and therefore does not contribute to the gradient of J˜W with respect to the model parameters.

7.5.1

Injecting Noise at the Output Targets

Most datasets have some amount of mistakes in the y labels. It can be harmful to maximize log p(y | x) when y is a mistake. One way to prevent this is to explicitly model the noise on the labels. For example, we can assume that for some small constant , the training set label y is correct with probability 1 − , and otherwise any of the other possible labels might be correct. This assumption is easy to incorporate into the cost function analytically, rather than by explicitly drawing noise samples. For example, label smoothing regularizes a model based on a softmax with k output values by replacing the hard 0 and 1 classiﬁcation targets with targets of k and 1− k−1 k , respectively. The standard cross-entropy loss may then be used with these soft targets. Maximum likelihood learning with a softmax classiﬁer and hard targets may actually never converge—the softmax can never predict a probability of exactly 0 or exactly 1, so it will continue to learn larger and larger weights, making more extreme predictions forever. It is possible to prevent this scenario using other regularization strategies like weight decay. Label smoothing has the advantage of preventing the pursuit of hard probabilities without discouraging correct classiﬁcation. This strategy has been used since the 1980s and continues to be featured prominently in modern neural networks (Szegedy et al., 2015).

7.6

Semi-Supervised Learning

In the paradigm of semi-supervised learning, both unlabeled examples from P (x) and labeled examples from P (x, y ) are used to estimate P (y | x) or predict y from 243

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x. In the context of deep learning, semi-supervised learning usually refers to learning a representation h = f (x). The goal is to learn a representation so that examples from the same class have similar representations. Unsupervised learning can provide useful cues for how to group examples in representation space. Examples that cluster tightly in the input space should be mapped to similar representations. A linear classiﬁer in the new space may achieve better generalization in many cases (Belkin and Niyogi, 2002; Chapelle et al., 2003). A long-standing variant of this approach is the application of principal components analysis as a pre-processing step before applying a classiﬁer (on the projected data). Instead of having separate unsupervised and supervised components in the model, one can construct models in which a generative model of either P (x) or P (x, y ) shares parameters with a discriminative model of P (y | x). One can then trade-oﬀ the supervised criterion − log P ( y | x) with the unsupervised or generative one (such as − log P (x) or − log P (x, y )). The generative criterion then expresses a particular form of prior belief about the solution to the supervised learning problem (Lasserre et al., 2006), namely that the structure of P (x) is connected to the structure of P (y | x ) in a way that is captured by the shared parametrization. By controlling how much of the generative criterion is included in the total criterion, one can ﬁnd a better trade-oﬀ than with a purely generative or a purely discriminative training criterion (Lasserre et al., 2006; Larochelle and Bengio, 2008). Salakhutdinov and Hinton (2008) describe a method for learning the kernel function of a kernel machine used for regression, in which the usage of unlabeled examples for modeling P (x) improves P (y | x) quite signiﬁcantly. See Chapelle et al. (2006) for more information about semi-supervised learning.

7.7

Multi-Task Learning

Multi-task learning (Caruana, 1993) is a way to improve generalization by pooling the examples (which can be seen as soft constraints imposed on the parameters) arising out of several tasks. In the same way that additional training examples put more pressure on the parameters of the model towards values that generalize well, when part of a model is shared across tasks, that part of the model is more constrained towards good values (assuming the sharing is justiﬁed), often yielding better generalization. 244

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Fig. 7.2 illustrates a very common form of multi-task learning, in which diﬀerent supervised tasks (predicting y(i) given x) share the same input x, as well as some intermediate-level representation h(shared) capturing a common pool of factors. The model can generally be divided into two kinds of parts and associated parameters: 1. Task-speciﬁc parameters (which only beneﬁt from the examples of their task to achieve good generalization). These are the upper layers of the neural network in Fig. 7.2. 2. Generic parameters, shared across all the tasks (which beneﬁt from the pooled data of all the tasks). These are the lower layers of the neural network in Fig. 7.2.

y (1)

y (2)

h(1)

h(2)

h(3)

h(shared)

x

Figure 7.2: Multi-task learning can be cast in several ways in deep learning frameworks and this ﬁgure illustrates the common situation where the tasks share a common input but involve diﬀerent target random variables. The lower layers of a deep network (whether it is supervised and feedforward or includes a generative component with downward arrows) can be shared across such tasks, while task-speciﬁc parameters (associated respectively with the weights into and from h(1) and h (2)) can be learned on top of those yielding a shared representation h(shared) . The underlying assumption is that there exists a common pool of factors that explain the variations in the input x, while each task is associated with a subset of these factors. In this example, it is additionally assumed that top-level hidden units h(1) and h(2) are specialized to each task (respectively predicting y(1) and y (2) ) while some intermediate-level representation h(shared) is shared across all tasks. In the unsupervised learning context, it makes sense for some of the top-level factors to be associated with none of the output tasks (h (3)): these are the factors that explain some of the input variations but are not relevant for predicting y(1) or y(2) . 245

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Figure 7.3: Learning curves showing how the negative log-likelihood loss changes over time (indicated as number of training iterations over the dataset, or epochs). In this example, we train a maxout network on MNIST. Observe that the training objective decreases consistently over time, but the validation set average loss eventually begins to increase again, forming an asymmetric U-shaped curve.

Improved generalization and generalization error bounds (Baxter, 1995) can be achieved because of the shared parameters, for which statistical strength can be greatly improved (in proportion with the increased number of examples for the shared parameters, compared to the scenario of single-task models). Of course this will happen only if some assumptions about the statistical relationship between the diﬀerent tasks are valid, meaning that there is something shared across some of the tasks. From the point of view of deep learning, the underlying prior belief is the following: among the factors that explain the variations observed in the data associated with the diﬀerent tasks, some are shared across two or more tasks.

7.8

Early Stopping

When training large models with suﬃcient representational capacity to overﬁt the task, we often observe that training error decreases steadily over time, but validation set error begins to rise again. See Fig. 7.3 for an example of this behavior. This behavior occurs very reliably. This means we can obtain a model with better validation set error (and thus, hopefully better test set error) by returning to the parameter setting at the point 246

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in time with the lowest validation set error. Instead of running our optimization algorithm until we reach a (local) minimum of validation error, we run it until the error on the validation set has not improved for some amount of time. Every time the error on the validation set improves, we store a copy of the model parameters. When the training algorithm terminates, we return these parameters, rather than the latest parameters. This procedure is speciﬁed more formally in Algorithm 7.1. Algorithm 7.1 The early stopping meta-algorithm for determining the best amount of time to train. This meta-algorithm is a general strategy that works well with a variety of training algorithms and ways of quantifying error on the validation set. Let n be the number of steps between evaluations. Let p be the “patience,” the number of times to observe worsening validation set error before giving up. Let θo be the initial parameters. θ ← θo i←0 j←0 v←∞ θ∗ ← θ i∗ ← i while j < p do Update θ by running the training algorithm for n steps. i← i+n v ← ValidationSetError(θ) if v < v then j←0 θ∗ ← θ i∗ ← i v ← v else j ←j+1 end if end while Best parameters are θ∗ , best number of training steps is i∗ This strategy is known as early stopping. It is probably the most commonly used form of regularization in deep learning. Its popularity is due both to its eﬀectiveness and its simplicity. 247

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One way to think of early stopping is as a very eﬃcient hyperparameter selection algorithm. In this view, the number of training steps is just another hyperparameter. We can see in Fig. 7.3 that this hyperparameter has a U-shaped validation set performance curve. Most hyperparameters that control model capacity have such a U-shaped validation set performance curve, as illustrated in Fig. 5.3. In the case of early stopping, we are controlling the eﬀective capacity of the model by determining how many steps it can take to ﬁt the training set. Most hyperparameters must be chosen using an expensive guess and check process, where we set a hyperparameter at the start of training, then run training for several steps to see its eﬀect. The “training time” hyperparameter is unique in that by deﬁnition a single run of training tries out many values of the hyperparameter. The only signiﬁcant cost to choosing this hyperparameter automatically via early stopping is running the validation set evaluation periodically during training. Ideally, this is done in parallel to the training process on a separate machine, separate CPU, or separate GPU from the main training process. If such resources are not available, then the cost of these periodic evaluations may be reduced by using a validation set that is small compared to the training set or by evaluating the validation set error less frequently and obtaining a lower resolution estimate of the optimal training time. An additional cost to early stopping is the need to maintain a copy of the best parameters. This cost is generally negligible, because it is acceptable to store these parameters in a slower and larger form of memory (for example, training in GPU memory, but storing the optimal parameters in host memory or on a disk drive). Since the best parameters are written to infrequently and never read during training, these occasional slow writes have little eﬀect on the total training time. Early stopping is a very unobtrusive form of regularization, in that it requires almost no change in the underlying training procedure, the objective function, or the set of allowable parameter values. This means that it is easy to use early stopping without damaging the learning dynamics. This is in contrast to weight decay, where one must be careful not to use too much weight decay and trap the network in a bad local minimum corresponding to a solution with pathologically small weights. Early stopping may be used either alone or in conjunction with other regularization strategies. Even when using regularization strategies that modify the objective function to encourage better generalization, it is rare for the best generalization to occur at a local minimum of the training objective. Early stopping requires a validation set, which means some training data is not fed to the model. To best exploit this extra data, one can perform extra training after the initial training with early stopping has completed. In the second, extra 248

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training step, all of the training data is included. There are two basic strategies one can use for this second training procedure. One strategy (Algorithm 7.2) is to initialize the model again and retrain on all of the data. In this second training pass, we train for the same number of steps as the early stopping procedure determined was optimal in the ﬁrst pass. There are some subtleties associated with this procedure. For example, there is not a good way of knowing whether to retrain for the same number of parameter updates or the same number of passes through the dataset. On the second round of training, each pass through the dataset will require more parameter updates because the training set is bigger. Algorithm 7.2 A meta-algorithm for using early stopping to determine how long to train, then retraining on all the data. Let X(train) and y(train) be the training set. Split X(train) and y(train) into (X (subtrain), X (valid)) and ( y(subtrain) , y (valid)) respectively. Run early stopping (Algorithm 7.1) starting from random θ using X(subtrain) and y(subtrain) for training data and X (valid) and y (valid) for validation data. This returns i∗, the optimal number of steps. Set θ to random values again. Train on X (train) and y(train) for i∗ steps. Another strategy for using all of the data is to keep the parameters obtained from the ﬁrst round of training and then continue training but now using all of the data. At this stage, we now no longer have a guide for when to stop in terms of a number of steps. Instead, we can monitor the average loss function on the validation set, and continue training until it falls below the value of the training set objective at which the early stopping procedure halted. This strategy avoids the high cost of retraining the model from scratch, but is not as well-behaved. For example, there is not any guarantee that the objective on the validation set will ever reach the target value, so this strategy is not even guaranteed to terminate. This procedure is presented more formally in Algorithm 7.3. Early stopping is also useful because it reduces the computational cost of the training procedure. Besides the obvious reduction in cost due to limiting the number of training iterations, it also has the beneﬁt of providing regularization without requiring the addition of penalty terms to the cost function or the computation of the gradients of such additional terms.

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Algorithm 7.3 Meta-algorithm using early stopping to determine at what objective value we start to overﬁt, then continue training until that value is reached. Let X(train) and y(train) be the training set. Split X(train) and y(train) into (X (subtrain), X (valid)) and ( y(subtrain) , y (valid)) respectively. Run early stopping (Algorithm 7.1) starting from random θ using X(subtrain) and y(subtrain) for training data and X (valid) and y (valid) for validation data. This updates θ. ← J (θ, X (subtrain) , y(subtrain)) while J (θ, X (valid) , y (valid) ) > do Train on X(train) and y(train) for n steps. end while How early stopping acts as a regularizer: So far we have stated that early stopping is a regularization strategy, but we have supported this claim only by showing learning curves where the validation set error has a U-shaped curve. What is the actual mechanism by which early stopping regularizes the model? Bishop (1995a) and Sjöberg and Ljung (1995) argued that early stopping has the eﬀect of restricting the optimization procedure to a relatively small volume of parameter space in the neighborhood of the initial parameter value θo . More speciﬁcally, imagine taking τ optimization steps (corresponding to τ training iterations) and with learning rate . We can view the product τ as a measure of eﬀective capacity. Assuming the gradient is bounded, restricting both the number of iterations and the learning rate limits the volume of parameter space reachable from θo . In this sense, τ behaves as if it were the reciprocal of the coeﬃcient used for weight decay. Indeed, we can show how—in the case of a simple linear model with a quadratic error function and simple gradient descent—early stopping is equivalent to L2 regularization. In order to compare with classical L2 regularization, we examine a simple setting where the only parameters are linear weights (θ = w). We can model the cost function J with a quadratic approximation in the neighborhood of the empirically optimal value of the weights w∗: 1 Jˆ(θ) = J (w∗) + (w − w ∗) H (w − w ∗ ), 2

(7.33)

where H is the Hessian matrix of J with respect to w evaluated at w∗ . Given the assumption that w∗ is a minimum of J(w), we know that H is positive semideﬁnite. 250

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w ˜

w∗ w2

w2

w∗

w1

w ˜

w1

Figure 7.4: An illustration of the eﬀect of early stopping. (Left) The solid contour lines indicate the contours of the negative log-likelihood. The dashed line indicates the trajectory taken by SGD beginning from the origin. Rather than stopping at the point w∗ that minimizes the cost, early stopping results in the trajectory stopping at an earlier ˜ (Right) An illustration of the eﬀect of L2 regularization for comparison. The point w. dashed circles indicate the contours of the L2 penalty, which causes the minimum of the total cost to lie nearer the origin than the minimum of the unregularized cost.

Under a local Taylor series approximation, the gradient is given by: ˆ w) = H (w − w ∗). ∇w J(

(7.34)

We are going to study the trajectory followed by the parameter vector during training. For simplicity, let us set the initial parameter vector to the origin,3 that is w(0) = 0. Let us suppose that we update the parameters via gradient descent: w (τ ) = w (τ−1) − ∇w J(w(τ−1) )

(7.35)

w(τ ) − w ∗ = (I − H )(w (τ−1) − w∗ )

(7.37)

= w (τ−1) − H (w(τ−1) − w ∗ )

(7.36)

Let us now rewrite this expression in the space of the eigenvectors of H , exploiting the eigendecomposition of H: H = QΛQ , where Λ is a diagonal matrix and Q is an orthonormal basis of eigenvectors. w (τ ) − w ∗ = (I − QΛQ )(w (τ−1) − w ∗ ) 3

(7.38)

For neural networks, to obtain symmetry breaking between hidden units, we cannot initialize all the parameters to 0, as discussed in Sec. 6.2. However, the argument holds for any other initial value w(0) . 251

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Q(w(τ ) − w ∗) = (I − Λ)Q (w(τ−1) − w∗ )

(7.39)

Assuming that w (0) = 0 and that is chosen to be small enough to guarantee |1 − λi | < 1, the parameter trajectory during training after τ parameter updates is as follows: Qw (τ ) = [I − (I − Λ)τ ]Q w∗ .

(7.40)

Q w ˜ = (Λ + αI )−1ΛQ w ∗

(7.41)

˜ = [I − (Λ + αI )−1 α]Qw ∗ Q w

(7.42)

˜ in Eq. 7.13 for L2 regularization can be rearranged Now, the expression for Q w as:

Comparing Eq. 7.40 and Eq. 7.42, we see that if the hyperparameters , α, and τ are chosen such that (7.43) (I − Λ)τ = (Λ + αI )−1 α,

then L2 regularization and early stopping can be seen to be equivalent (at least under the quadratic approximation of the objective function). Going even further, by taking logarithms and using the series expansion for log(1 + x ), we can conclude that if all λi are small (that is, λi 1 and λi/α 1) then 1 (7.44) τ≈ , α 1 (7.45) α≈ . τ That is, under these assumptions, the number of training iterations τ plays a role inversely proportional to the L2 regularization parameter, and the inverse of τ plays the role of the weight decay coeﬃcient. Parameter values corresponding to directions of signiﬁcant curvature (of the objective function) are regularized less than directions of less curvature. Of course, in the context of early stopping, this really means that parameters that correspond to directions of signiﬁcant curvature tend to learn early relative to parameters corresponding to directions of less curvature. The derivations in this section have shown that a trajectory of length τ ends at a point that corresponds to a minimum of the L2-regularized objective. Early stopping is of course more than the mere restriction of the trajectory length; instead, early stopping typically involves monitoring the validation set error in order to stop the trajectory at a particularly good point in space. Early stopping therefore has the advantage over weight decay that early stopping automatically determines the correct amount of regularization while weight decay requires many training experiments with diﬀerent values of its hyperparameter. 252

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7.9

Parameter Tying and Parameter Sharing

Thus far, in this chapter, when we have discussed adding constraints or penalties to the parameters, we have always done so with respect to a ﬁxed region or point. For example, L2 regularization (or weight decay) penalizes model parameters for deviating from the ﬁxed value of zero. However, sometimes we may need other ways to express our prior knowledge about suitable values of the model parameters. Sometimes we might not know precisely what values the parameters should take but we know, from knowledge of the domain and model architecture, that there should be some dependencies between the model parameters. A common type of dependency that we often want to express is that certain parameters should be close to one another. Consider the following scenario: we have two models performing the same classiﬁcation task (with the same set of classes) but with somewhat diﬀerent input distributions. Formally, we have model A with parameters w (A) and model B with parameters w(B) . The two models map the input to two diﬀerent, but related outputs: yˆ (A) = f(w(A) , x) and yˆ (B) = g(w (B), x). Let us imagine that the tasks are similar enough (perhaps with similar input and output distributions) that we believe the model parameters should be close to each other: ∀i, w i(A) should be close to w(i B) . We can leverage this information through regularization. Speciﬁcally, we can use a parameter norm penalty of the form: Ω(w (A), w (B)) = w (A) − w(B)22. Here we used an L2 penalty, but other choices are also possible. This kind of approach was proposed by Lasserre et al. (2006), who regularized the parameters of one model, trained as a classiﬁer in a supervised paradigm, to be close to the parameters of another model, trained in an unsupervised paradigm (to capture the distribution of the observed input data). The architectures were constructed such that many of the parameters in the classiﬁer model could be paired to corresponding parameters in the unsupervised model. While a parameter norm penalty is one way to regularize parameters to be close to one another, the more popular way is to use constraints: to force sets of parameters to be equal. This method of regularization is often referred to as parameter sharing, where we interpret the various models or model components as sharing a unique set of parameters. A signiﬁcant advantage of parameter sharing over regularizing the parameters to be close (via a norm penalty) is that only a subset of the parameters (the unique set) need to be stored in memory. In certain models—such as the convolutional neural network—this can lead to signiﬁcant reduction in the memory footprint of the model. 253

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Convolutional Neural Networks By far the most popular and extensive use of parameter sharing occurs in convolutional neural networks (CNNs) applied to computer vision. Natural images have many statistical properties that are invariant to translation. For example, a photo of a cat remains a photo of a cat if it is translated one pixel to the right. CNNs take this property into account by sharing parameters across multiple image locations. The same feature (a hidden unit with the same weights) is computed over diﬀerent locations in the input. This means that we can ﬁnd a cat with the same cat detector whether the cat appears at column i or column i + 1 in the image. Parameter sharing has allowed CNNs to dramatically lower the number of unique model parameters and to signiﬁcantly increase network sizes without requiring a corresponding increase in training data. It remains one of the best examples of how to eﬀectively incorporate domain knowledge into the network architecture. CNNs will be discussed in more detail in Chapter 9.

7.10

Sparse Representations

Weight decay acts by placing a penalty directly on the model parameters. Another strategy is to place a penalty on the activations of the units in a neural network, encouraging their activations to be sparse. This indirectly imposes a complicated penalty on the model parameters. We have already discussed (in Sec. 7.1.2) how L1 penalization induces a sparse parametrization—meaning that many of the parameters become zero (or close to zero). Representational sparsity, on the other hand, describes a representation where many of the elements of the representation are zero (or close to zero). A simpliﬁed view of this distinction can be illustrated in the context of linear regression: 2 18 4 0 0 −2 0 0 5 0 0 −1 0 3 3 0 −2 15 = 0 5 0 0 0 0 −5 (7.46) −9 1 0 0 −1 0 −4 1 −3 1 0 0 0 −5 0 4 m × n m y∈R A∈R x ∈ Rn

254

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−14 1 19 2 23

=

y ∈ Rm

3 4 −1 3 −5

−1 2 −5 4 1 2 −3 −1 1 3 5 4 2 −3 −2 1 2 −3 0 −3 4 −2 2 −5 −1 B ∈ R m× n

0 2 0 0 −3 0 h ∈ Rn

(7.47)

In the ﬁrst expression, we have an example of a sparsely parametrized linear regression model. In the second, we have linear regression with a sparse representation h of the data x. That is, h is a function of x that, in some sense, represents the information present in x, but does so with a sparse vector. Representational regularization is accomplished by the same sorts of mechanisms that we have used in parameter regularization. Norm penalty regularization of representations is performed by adding to the loss function J a norm penalty on the representation. This penalty is denoted ˜ Ω(h). As before, we denote the regularized loss function by J: J˜(θ; X , y) = J (θ; X , y) + αΩ(h)

(7.48)

where α ∈ [0, ∞) weights the relative contribution of the norm penalty term, with larger values of α corresponding to more regularization. Just as an L1 penalty on the parameters induces parameter sparsity, an L1 penalty on the elements of the representation induces representational sparsity: Ω(h) = ||h||1 = i |h i|. Of course, the L 1 penalty is only one choice of penalty that can result in a sparse representation. Others include the penalty derived from a Student-t prior on the representation (Olshausen and Field, 1996; Bergstra, 2011) and KL divergence penalties (Larochelle and Bengio, 2008) that are especially useful for representations with elements constrained to lie on the unit interval. Lee et al. (2008) and Goodfellow et al. (2009) both provide examples of strategies 1 based on regularizing the average activation across several examples, m i h(i), to be near some target value, such as a vector with .01 for each entry.

Other approaches obtain representational sparsity with a hard constraint on the activation values. For example, orthogonal matching pursuit (Pati et al., 1993) encodes an input x with the representation h that solves the constrained optimization problem arg min x − W h2 , (7.49) h,h 0 2 thus seems hopeless. What can we do instead? Batch normalization provides an elegant way of reparametrizing almost any deep network. The reparametrization signiﬁcantly reduces the problem of coordinating updates across many layers. Batch normalization can be applied to any input or hidden layer in a network. Let H be a minibatch of activations of the layer to normalize, arranged as a design matrix, with the activations for each example appearing in a row of the matrix. To normalize H , we replace it with H =

H−µ , σ

(8.36)

where µ is a vector containing the mean of each unit and σ is a vector containing the standard deviation of each unit. The arithmetic here is based on broadcasting the vector µ and the vector σ to be applied to every row of the matrix H. Within each row, the arithmetic is element-wise, so Hi,j is normalized by subtracting µj and dividing by σj . The rest of the network then operates on H in exactly the same way that the original network operated on H . At training time, µ=

1 Hi,: m i 318

(8.37)

CHAPTER 8. OPTIMIZATION FOR TRAINING DEEP MODELS

and σ=

δ+

1 (H − µ)2i , m i

(8.38)

where δ is a small positive value such as 10 −8 imposed to avoid encountering the √ undeﬁned gradient of z at z = 0. Crucially, we back-propagate through these operations for computing the mean and the standard deviation, and for applying them to normalize H . This means that the gradient will never propose an operation that acts simply to increase the standard deviation or mean of hi ; the normalization operations remove the eﬀect of such an action and zero out its component in the gradient. This was a major innovation of the batch normalization approach. Previous approaches had involved adding penalties to the cost function to encourage units to have normalized activation statistics or involved intervening to renormalize unit statistics after each gradient descent step. The former approach usually resulted in imperfect normalization and the latter usually resulted in signiﬁcant wasted time as the learning algorithm repeatedly proposed changing the mean and variance and the normalization step repeatedly undid this change. Batch normalization reparametrizes the model to make some units always be standardized by deﬁnition, deftly sidestepping both problems. At test time, µ and σ may be replaced by running averages that were collected during training time. This allows the model to be evaluated on a single example, without needing to use deﬁnitions of µ and σ that depend on an entire minibatch. Revisiting the yˆ = xw1w2 . . . wl example, we see that we can mostly resolve the diﬃculties in learning this model by normalizing hl−1 . Suppose that x is drawn from a unit Gaussian. Then hl−1 will also come from a Gaussian, because the transformation from x to hl is linear. However, h l−1 will no longer have zero mean and unit variance. After applying batch normalization, we obtain the normalized ˆhl−1 that restores the zero mean and unit variance properties. For almost any update to the lower layers, ˆhl−1 will remain a unit Gaussian. The output yˆ may then be learned as a simple linear function yˆ = wl hˆ l−1. Learning in this model is now very simple because the parameters at the lower layers simply do not have an eﬀect in most cases; their output is always renormalized to a unit Gaussian. In some corner cases, the lower layers can have an eﬀect. Changing one of the lower layer weights to 0 can make the output become degenerate, and changing the sign of one of the lower weights can ﬂip the relationship between hˆl−1 and y. These situations are very rare. Without normalization, nearly every update would have an extreme eﬀect on the statistics of hl−1 . Batch normalization has thus made this model signiﬁcantly easier to learn. In this example, the ease of learning of course came at the cost of making the lower layers useless. In our linear example, 319

CHAPTER 8. OPTIMIZATION FOR TRAINING DEEP MODELS

the lower layers no longer have any harmful eﬀect, but they also no longer have any beneﬁcial eﬀect. This is because we have normalized out the ﬁrst and second order statistics, which is all that a linear network can inﬂuence. In a deep neural network with nonlinear activation functions, the lower layers can perform nonlinear transformations of the data, so they remain useful. Batch normalization acts to standardize only the mean and variance of each unit in order to stabilize learning, but allows the relationships between units and the nonlinear statistics of a single unit to change. Because the ﬁnal layer of the network is able to learn a linear transformation, we may actually wish to remove all linear relationships between units within a layer. Indeed, this is the approach taken by Desjardins et al. (2015), who provided the inspiration for batch normalization. Unfortunately, eliminating all linear interactions is much more expensive than standardizing the mean and standard deviation of each individual unit, and so far batch normalization remains the most practical approach. Normalizing the mean and standard deviation of a unit can reduce the expressive power of the neural network containing that unit. In order to maintain the expressive power of the network, it is common to replace the batch of hidden unit activations H with γH +β rather than simply the normalized H . The variables γ and β are learned parameters that allow the new variable to have any mean and standard deviation. At ﬁrst glance, this may seem useless—why did we set the mean to 0 , and then introduce a parameter that allows it to be set back to any arbitrary value β? The answer is that the new parametrization can represent the same family of functions of the input as the old parametrization, but the new parametrization has diﬀerent learning dynamics. In the old parametrization, the mean of H was determined by a complicated interaction between the parameters in the layers below H. In the new parametrization, the mean of γH + β is determined solely by β. The new parametrization is much easier to learn with gradient descent. Most neural network layers take the form of φ(XW + b) where φ is some ﬁxed nonlinear activation function such as the rectiﬁed linear transformation. It is natural to wonder whether we should apply batch normalization to the input X, or to the transformed value XW + b. Ioﬀe and Szegedy (2015) recommend the latter. More speciﬁcally, XW + b should be replaced by a normalized version of XW . The bias term should be omitted because it becomes redundant with the β parameter applied by the batch normalization reparametrization. The input to a layer is usually the output of a nonlinear activation function such as the rectiﬁed linear function in a previous layer. The statistics of the input are thus 320

CHAPTER 8. OPTIMIZATION FOR TRAINING DEEP MODELS

more non-Gaussian and less amenable to standardization by linear operations. In convolutional networks, described in Chapter 9, it is important to apply the same normalizing µ and σ at every spatial location within a feature map, so that the statistics of the feature map remain the same regardless of spatial location.

8.7.2

Coordinate Descent

In some cases, it may be possible to solve an optimization problem quickly by breaking it into separate pieces. If we minimize f (x) with respect to a single variable xi , then minimize it with respect to another variable xj and so on, repeatedly cycling through all variables, we are guaranteed to arrive at a (local) minimum. This practice is known as coordinate descent, because we optimize one coordinate at a time. More generally, block coordinate descent refers to minimizing with respect to a subset of the variables simultaneously. The term “coordinate descent” is often used to refer to block coordinate descent as well as the strictly individual coordinate descent. Coordinate descent makes the most sense when the diﬀerent variables in the optimization problem can be clearly separated into groups that play relatively isolated roles, or when optimization with respect to one group of variables is signiﬁcantly more eﬃcient than optimization with respect to all of the variables. For example, consider the cost function 2 J (H , W ) = . (8.39) |Hi,j | + X−W H i,j

i,j

i,j

This function describes a learning problem called sparse coding, where the goal is to ﬁnd a weight matrix W that can linearly decode a matrix of activation values H to reconstruct the training set X. Most applications of sparse coding also involve weight decay or a constraint on the norms of the columns of W , in order to prevent the pathological solution with extremely small H and large W . The function J is not convex. However, we can divide the inputs to the training algorithm into two sets: the dictionary parameters W and the code representations H . Minimizing the objective function with respect to either one of these sets of variables is a convex problem. Block coordinate descent thus gives us an optimization strategy that allows us to use eﬃcient convex optimization algorithms, by alternating between optimizing W with H ﬁxed, then optimizing H with W ﬁxed. Coordinate descent is not a very good strategy when the value of one variable strongly inﬂuences the optimal value of another variable, as in the function f (x) = 321

CHAPTER 8. OPTIMIZATION FOR TRAINING DEEP MODELS

(x1 − x2)2 + α x 21 + x 22 where α is a positive constant. The ﬁrst term encourages the two variables to have similar value, while the second term encourages them to be near zero. The solution is to set both to zero. Newton’s method can solve the problem in a single step because it is a positive deﬁnite quadratic problem. However, for small α, coordinate descent will make very slow progress because the ﬁrst term does not allow a single variable to be changed to a value that diﬀers signiﬁcantly from the current value of the other variable.

8.7.3

Polyak Averaging

Polyak averaging (Polyak and Juditsky, 1992) consists of averaging together several points in the trajectory through parameter space visited by an optimization θ(1) , . . . , θ (t), then the algorithm. If t iterations of gradient descent visit points (i) output of the Polyak averaging algorithm is θˆ(t) = 1t i θ . On some problem classes, such as gradient descent applied to convex problems, this approach has strong convergence guarantees. When applied to neural networks, its justiﬁcation is more heuristic, but it performs well in practice. The basic idea is that the optimization algorithm may leap back and forth across a valley several times without ever visiting a point near the bottom of the valley. The average of all of the locations on either side should be close to the bottom of the valley though. In non-convex problems, the path taken by the optimization trajectory can be very complicated and visit many diﬀerent regions. Including points in parameter space from the distant past that may be separated from the current point by large barriers in the cost function does not seem like a useful behavior. As a result, when applying Polyak averaging to non-convex problems, it is typical to use an exponentially decaying running average: θˆ(t) = αθˆ(t−1) + (1 − α)θ (t).

(8.40)

The running average approach is used in numerous applications. See Szegedy et al. (2015) for a recent example.

8.7.4

Supervised Pretraining

Sometimes, directly training a model to solve a speciﬁc task can be too ambitious if the model is complex and hard to optimize or if the task is very diﬃcult. It is sometimes more eﬀective to train a simpler model to solve the task, then make the model more complex. It can also be more eﬀective to train the model to solve a simpler task, then move on to confront the ﬁnal task. These strategies that involve 322

CHAPTER 8. OPTIMIZATION FOR TRAINING DEEP MODELS

training simple models on simple tasks before confronting the challenge of training the desired model to perform the desired task are collectively known as pretraining. Greedy algorithms break a problem into many components, then solve for the optimal version of each component in isolation. Unfortunately, combining the individually optimal components is not guaranteed to yield an optimal complete solution. However, greedy algorithms can be computationally much cheaper than algorithms that solve for the best joint solution, and the quality of a greedy solution is often acceptable if not optimal. Greedy algorithms may also be followed by a ﬁne-tuning stage in which a joint optimization algorithm searches for an optimal solution to the full problem. Initializing the joint optimization algorithm with a greedy solution can greatly speed it up and improve the quality of the solution it ﬁnds. Pretraining, and especially greedy pretraining, algorithms are ubiquitous in deep learning. In this section, we describe speciﬁcally those pretraining algorithms that break supervised learning problems into other simpler supervised learning problems. This approach is known as greedy supervised pretraining. In the original (Bengio et al., 2007) version of greedy supervised pretraining, each stage consists of a supervised learning training task involving only a subset of the layers in the ﬁnal neural network. An example of greedy supervised pretraining is illustrated in Fig. 8.7, in which each added hidden layer is pretrained as part of a shallow supervised MLP, taking as input the output of the previously trained hidden layer. Instead of pretraining one layer at a time, Simonyan and Zisserman (2015) pretrain a deep convolutional network (eleven weight layers) and then use the ﬁrst four and last three layers from this network to initialize even deeper networks (with up to nineteen layers of weights). The middle layers of the new, very deep network are initialized randomly. The new network is then jointly trained. Another option, explored by Yu et al. (2010) is to use the outputs of the previously trained MLPs, as well as the raw input, as inputs for each added stage. Why would greedy supervised pretraining help? The hypothesis initially discussed by Bengio et al. (2007) is that it helps to provide better guidance to the intermediate levels of a deep hierarchy. In general, pretraining may help both in terms of optimization and in terms of generalization. An approach related to supervised pretraining extends the idea to the context of transfer learning: Yosinski et al. (2014) pretrain a deep convolutional net with 8 layers of weights on a set of tasks (a subset of the 1000 ImageNet object categories) and then initialize a same-size network with the ﬁrst k layers of the ﬁrst net. All the layers of the second network (with the upper layers initialized randomly) are 323

CHAPTER 8. OPTIMIZATION FOR TRAINING DEEP MODELS

y

U(1) h(1)

h(1)

W (1)

W (1) x

x

(a)

(b)

U (1)

y

U (1)

y

y U (2) h(2) W (2)

h(2) U(2)

W (2)

y

h(1)

h(1) U(1)

W (1)

y

W (1)

x

x

(c)

(d)

Figure 8.7: Illustration of one form of greedy supervised pretraining (Bengio et al., 2007). (a) We start by training a suﬃciently shallow architecture. (b) Another drawing of the same architecture. (c) We keep only the input-to-hidden layer of the original network and discard the hidden-to-output layer. We send the output of the ﬁrst hidden layer as input to another supervised single hidden layer MLP that is trained with the same objective as the ﬁrst network was, thus adding a second hidden layer. This can be repeated for as many layers as desired. (d) Another drawing of the result, viewed as a feedforward network. To further improve the optimization, we can jointly ﬁne-tune all the layers, either only at the end or at each stage of this process.

324

CHAPTER 8. OPTIMIZATION FOR TRAINING DEEP MODELS

then jointly trained to perform a diﬀerent set of tasks (another subset of the 1000 ImageNet object categories), with fewer training examples than for the ﬁrst set of tasks. Other approaches to transfer learning with neural networks are discussed in Sec. 15.2. Another related line of work is the FitNets (Romero et al., 2015) approach. This approach begins by training a network that has low enough depth and great enough width (number of units per layer) to be easy to train. This network then becomes a teacher for a second network, designated the student. The student network is much deeper and thinner (eleven to nineteen layers) and would be diﬃcult to train with SGD under normal circumstances. The training of the student network is made easier by training the student network not only to predict the output for the original task, but also to predict the value of the middle layer of the teacher network. This extra task provides a set of hints about how the hidden layers should be used and can simplify the optimization problem. Additional parameters are introduced to regress the middle layer of the 5-layer teacher network from the middle layer of the deeper student network. However, instead of predicting the ﬁnal classiﬁcation target, the objective is to predict the middle hidden layer of the teacher network. The lower layers of the student networks thus have two objectives: to help the outputs of the student network accomplish their task, as well as to predict the intermediate layer of the teacher network. Although a thin and deep network appears to be more diﬃcult to train than a wide and shallow network, the thin and deep network may generalize better and certainly has lower computational cost if it is thin enough to have far fewer parameters. Without the hints on the hidden layer, the student network performs very poorly in the experiments, both on the training and test set. Hints on middle layers may thus be one of the tools to help train neural networks that otherwise seem diﬃcult to train, but other optimization techniques or changes in the architecture may also solve the problem.

8.7.5

Designing Models to Aid Optimization

To improve optimization, the best strategy is not always to improve the optimization algorithm. Instead, many improvements in the optimization of deep models have come from designing the models to be easier to optimize. In principle, we could use activation functions that increase and decrease in jagged non-monotonic patterns. However, this would make optimization extremely diﬃcult. In practice, it is more important to choose a model family that is easy to optimize than to use a powerful optimization algorithm. Most of the advances in neural network learning over the past 30 years have been 325

CHAPTER 8. OPTIMIZATION FOR TRAINING DEEP MODELS

obtained by changing the model family rather than changing the optimization procedure. Stochastic gradient descent with momentum, which was used to train neural networks in the 1980s, remains in use in modern state of the art neural network applications. Speciﬁcally, modern neural networks reﬂect a design choice to use linear transformations between layers and activation functions that are diﬀerentiable almost everywhere and have signiﬁcant slope in large portions of their domain. In particular, model innovations like the LSTM, rectiﬁed linear units and maxout units have all moved toward using more linear functions than previous models like deep networks based on sigmoidal units. These models have nice properties that make optimization easier. The gradient ﬂows through many layers provided that the Jacobian of the linear transformation has reasonable singular values. Moreover, linear functions consistently increase in a single direction, so even if the model’s output is very far from correct, it is clear simply from computing the gradient which direction its output should move to reduce the loss function. In other words, modern neural nets have been designed so that their local gradient information corresponds reasonably well to moving toward a distant solution. Other model design strategies can help to make optimization easier. For example, linear paths or skip connections between layers reduce the length of the shortest path from the lower layer’s parameters to the output, and thus mitigate the vanishing gradient problem (Srivastava et al., 2015). A related idea to skip connections is adding extra copies of the output that are attached to the intermediate hidden layers of the network, as in GoogLeNet (Szegedy et al., 2014a) and deeply-supervised nets (Lee et al., 2014). These “auxiliary heads” are trained to perform the same task as the primary output at the top of the network in order to ensure that the lower layers receive a large gradient. When training is complete the auxiliary heads may be discarded. This is an alternative to the pretraining strategies, which were introduced in the previous section. In this way, one can train jointly all the layers in a single phase but change the architecture, so that intermediate layers (especially the lower ones) can get some hints about what they should do, via a shorter path. These hints provide an error signal to lower layers.

8.7.6

Continuation Methods and Curriculum Learning

As argued in Sec. 8.2.7, many of the challenges in optimization arise from the global structure of the cost function and cannot be resolved merely by making better estimates of local update directions. The predominant strategy for overcoming this problem is to attempt to initialize the parameters in a region that is connected to the solution by a short path through parameter space that local descent can 326

CHAPTER 8. OPTIMIZATION FOR TRAINING DEEP MODELS

discover. Continuation methods are a family of strategies that can make optimization easier by choosing initial points to ensure that local optimization spends most of its time in well-behaved regions of space. The idea behind continuation methods is to construct a series of objective functions over the same parameters. In order to minimize a cost function J (θ ), we will construct new cost functions {J (0) , . . . , J (n)}. These cost functions are designed to be increasingly diﬃcult, with J (0) being fairly easy to minimize, and J (n) , the most diﬃcult, being J (θ), the true cost function motivating the entire process. When we say that J (i) is easier than J (i+1), we mean that it is well behaved over more of θ space. A random initialization is more likely to land in the region where local descent can minimize the cost function successfully because this region is larger. The series of cost functions are designed so that a solution to one is a good initial point of the next. We thus begin by solving an easy problem then reﬁne the solution to solve incrementally harder problems until we arrive at a solution to the true underlying problem. Traditional continuation methods (predating the use of continuation methods for neural network training) are usually based on smoothing the objective function. See Wu (1997) for an example of such a method and a review of some related methods. Continuation methods are also closely related to simulated annealing, which adds noise to the parameters (Kirkpatrick et al., 1983). Continuation methods have been extremely successful in recent years. See Mobahi and Fisher (2015) for an overview of recent literature, especially for AI applications. Continuation methods traditionally were mostly designed with the goal of overcoming the challenge of local minima. Speciﬁcally, they were designed to reach a global minimum despite the presence of many local minima. To do so, these continuation methods would construct easier cost functions by “blurring” the original cost function. This blurring operation can be done by approximating J (i)(θ) = E θ ∼N (θ ;θ,σ (i)2 )J(θ )

(8.41)

via sampling. The intuition for this approach is that some non-convex functions become approximately convex when blurred. In many cases, this blurring preserves enough information about the location of a global minimum that we can ﬁnd the global minimum by solving progressively less blurred versions of the problem. This approach can break down in three diﬀerent ways. First, it might successfully deﬁne a series of cost functions where the ﬁrst is convex and the optimum tracks from one function to the next arriving at the global minimum, but it might require so many incremental cost functions that the cost of the entire procedure remains high. NP-hard optimization problems remain NP-hard, even when continuation methods 327

CHAPTER 8. OPTIMIZATION FOR TRAINING DEEP MODELS

are applicable. The other two ways that continuation methods fail both correspond to the method not being applicable. First, the function might not become convex, no matter how much it is blurred. Consider for example the function J(θ) = −θ θ. Second, the function may become convex as a result of blurring, but the minimum of this blurred function may track to a local rather than a global minimum of the original cost function. Though continuation methods were mostly originally designed to deal with the problem of local minima, local minima are no longer believed to be the primary problem for neural network optimization. Fortunately, continuation methods can still help. The easier objective functions introduced by the continuation method can eliminate ﬂat regions, decrease variance in gradient estimates, improve conditioning of the Hessian matrix, or do anything else that will either make local updates easier to compute or improve the correspondence between local update directions and progress toward a global solution. Bengio et al. (2009) observed that an approach called curriculum learning or shaping can be interpreted as a continuation method. Curriculum learning is based on the idea of planning a learning process to begin by learning simple concepts and progress to learning more complex concepts that depend on these simpler concepts. This basic strategy was previously known to accelerate progress in animal training (Skinner, 1958; Peterson, 2004; Krueger and Dayan, 2009) and machine learning (Solomonoﬀ, 1989; Elman, 1993; Sanger, 1994). Bengio et al. (2009) justiﬁed this strategy as a continuation method, where earlier J (i) are made easier by increasing the inﬂuence of simpler examples (either by assigning their contributions to the cost function larger coeﬃcients, or by sampling them more frequently), and experimentally demonstrated that better results could be obtained by following a curriculum on a large-scale neural language modeling task. Curriculum learning has been successful on a wide range of natural language (Spitkovsky et al., 2010; Collobert et al., 2011a; Mikolov et al., 2011b; Tu and Honavar, 2011) and computer vision (Kumar et al., 2010; Lee and Grauman, 2011; Supancic and Ramanan, 2013) tasks. Curriculum learning was also veriﬁed as being consistent with the way in which humans teach (Khan et al., 2011): teachers start by showing easier and more prototypical examples and then help the learner reﬁne the decision surface with the less obvious cases. Curriculum-based strategies are more eﬀective for teaching humans than strategies based on uniform sampling of examples, and can also increase the eﬀectiveness of other teaching strategies (Basu and Christensen, 2013). Another important contribution to research on curriculum learning arose in the context of training recurrent neural networks to capture long-term dependencies: 328

CHAPTER 8. OPTIMIZATION FOR TRAINING DEEP MODELS

Zaremba and Sutskever (2014) found that much better results were obtained with a stochastic curriculum, in which a random mix of easy and diﬃcult examples is always presented to the learner, but where the average proportion of the more diﬃcult examples (here, those with longer-term dependencies) is gradually increased. With a deterministic curriculum, no improvement over the baseline (ordinary training from the full training set) was observed. We have now described the basic family of neural network models and how to regularize and optimize them. In the chapters ahead, we turn to specializations of the neural network family, that allow neural networks to scale to very large sizes and process input data that has special structure. The optimization methods discussed in this chapter are often directly applicable to these specialized architectures with little or no modiﬁcation.

329

Chapter 9

Convolutional Networks Convolutional networks (LeCun, 1989), also known as convolutional neural networks or CNNs, are a specialized kind of neural network for processing data that has a known, grid-like topology. Examples include time-series data, which can be thought of as a 1D grid taking samples at regular time intervals, and image data, which can be thought of as a 2D grid of pixels. Convolutional networks have been tremendously successful in practical applications. The name “convolutional neural network” indicates that the network employs a mathematical operation called convolution. Convolution is a specialized kind of linear operation. Convolutional networks are simply neural networks that use convolution in place of general matrix multiplication in at least one of their layers. In this chapter, we will ﬁrst describe what convolution is. Next, we will explain the motivation behind using convolution in a neural network. We will then describe an operation called pooling, which almost all convolutional networks employ. Usually, the operation used in a convolutional neural network does not correspond precisely to the deﬁnition of convolution as used in other ﬁelds such as engineering or pure mathematics. We will describe several variants on the convolution function that are widely used in practice for neural networks. We will also show how convolution may be applied to many kinds of data, with diﬀerent numbers of dimensions. We then discuss means of making convolution more eﬃcient. Convolutional networks stand out as an example of neuroscientiﬁc principles inﬂuencing deep learning. We will discuss these neuroscientiﬁc principles, then conclude with comments about the role convolutional networks have played in the history of deep learning. One topic this chapter does not address is how to choose the architecture of your convolutional network. The goal of this chapter is to describe the kinds of tools that convolutional networks provide, while Chapter 11 330

CHAPTER 9. CONVOLUTIONAL NETWORKS

describes general guidelines for choosing which tools to use in which circumstances. Research into convolutional network architectures proceeds so rapidly that a new best architecture for a given benchmark is announced every few weeks to months, rendering it impractical to describe the best architecture in print. However, the best architectures have consistently been composed of the building blocks described here.

9.1

The Convolution Operation

In its most general form, convolution is an operation on two functions of a realvalued argument. To motivate the deﬁnition of convolution, we start with examples of two functions we might use. Suppose we are tracking the location of a spaceship with a laser sensor. Our laser sensor provides a single output x(t), the position of the spaceship at time t. Both x and t are real-valued, i.e., we can get a diﬀerent reading from the laser sensor at any instant in time. Now suppose that our laser sensor is somewhat noisy. To obtain a less noisy estimate of the spaceship’s position, we would like to average together several measurements. Of course, more recent measurements are more relevant, so we will want this to be a weighted average that gives more weight to recent measurements. We can do this with a weighting function w(a), where a is the age of a measurement. If we apply such a weighted average operation at every moment, we obtain a new function s providing a smoothed estimate of the position of the spaceship: s(t) = x(a)w (t − a)da (9.1) This operation is called convolution. The convolution operation is typically denoted with an asterisk: s(t) = (x ∗ w )(t) (9.2) In our example, w needs to be a valid probability density function, or the output is not a weighted average. Also, w needs to be 0 for all negative arguments, or it will look into the future, which is presumably beyond our capabilities. These limitations are particular to our example though. In general, convolution is deﬁned for any functions for which the above integral is deﬁned, and may be used for other purposes besides taking weighted averages. In convolutional network terminology, the ﬁrst argument (in this example, the function x) to the convolution is often referred to as the input and the second 331

CHAPTER 9. CONVOLUTIONAL NETWORKS

argument (in this example, the function w) as the kernel. The output is sometimes referred to as the feature map. In our example, the idea of a laser sensor that can provide measurements at every instant in time is not realistic. Usually, when we work with data on a computer, time will be discretized, and our sensor will provide data at regular intervals. In our example, it might be more realistic to assume that our laser provides a measurement once per second. The time index t can then take on only integer values. If we now assume that x and w are deﬁned only on integer t, we can deﬁne the discrete convolution: s(t) = (x ∗ w )(t) =

∞

a=−∞

x(a)w (t − a)

(9.3)

In machine learning applications, the input is usually a multidimensional array of data and the kernel is usually a multidimensional array of parameters that are adapted by the learning algorithm. We will refer to these multidimensional arrays as tensors. Because each element of the input and kernel must be explicitly stored separately, we usually assume that these functions are zero everywhere but the ﬁnite set of points for which we store the values. This means that in practice we can implement the inﬁnite summation as a summation over a ﬁnite number of array elements. Finally, we often use convolutions over more than one axis at a time. For example, if we use a two-dimensional image I as our input, we probably also want to use a two-dimensional kernel K : S (i, j ) = (I ∗ K )(i, j ) = I (m, n)K (i − m, j − n). (9.4) m

n

Convolution is commutative, meaning we can equivalently write: S (i, j ) = (K ∗ I )(i, j ) = I (i − m, j − n)K (m, n). m

(9.5)

n

Usually the latter formula is more straightforward to implement in a machine learning library, because there is less variation in the range of valid values of m and n. The commutative property of convolution arises because we have ﬂipped the kernel relative to the input, in the sense that as m increases, the index into the input increases, but the index into the kernel decreases. The only reason to ﬂip the kernel is to obtain the commutative property. While the commutative property 332

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is useful for writing proofs, it is not usually an important property of a neural network implementation. Instead, many neural network libraries implement a related function called the cross-correlation, which is the same as convolution but without ﬂipping the kernel: I (i + m, j + n)K (m, n). (9.6) S (i, j ) = (I ∗ K )(i, j ) = m

n

Many machine learning libraries implement cross-correlation but call it convolution. In this text we will follow this convention of calling both operations convolution, and specify whether we mean to ﬂip the kernel or not in contexts where kernel ﬂipping is relevant. In the context of machine learning, the learning algorithm will learn the appropriate values of the kernel in the appropriate place, so an algorithm based on convolution with kernel ﬂipping will learn a kernel that is ﬂipped relative to the kernel learned by an algorithm without the ﬂipping. It is also rare for convolution to be used alone in machine learning; instead convolution is used simultaneously with other functions, and the combination of these functions does not commute regardless of whether the convolution operation ﬂips its kernel or not. See Fig. 9.1 for an example of convolution (without kernel ﬂipping) applied to a 2-D tensor. Discrete convolution can be viewed as multiplication by a matrix. However, the matrix has several entries constrained to be equal to other entries. For example, for univariate discrete convolution, each row of the matrix is constrained to be equal to the row above shifted by one element. This is known as a Toeplitz matrix. In two dimensions, a doubly block circulant matrix corresponds to convolution. In addition to these constraints that several elements be equal to each other, convolution usually corresponds to a very sparse matrix (a matrix whose entries are mostly equal to zero). This is because the kernel is usually much smaller than the input image. Any neural network algorithm that works with matrix multiplication and does not depend on speciﬁc properties of the matrix structure should work with convolution, without requiring any further changes to the neural network. Typical convolutional neural networks do make use of further specializations in order to deal with large inputs eﬃciently, but these are not strictly necessary from a theoretical perspective.

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9.2

Motivation

Convolution leverages three important ideas that can help improve a machine learning system: sparse interactions, parameter sharing and equivariant representations. Moreover, convolution provides a means for working with inputs of variable size. We now describe each of these ideas in turn. Traditional neural network layers use matrix multiplication by a matrix of parameters with a separate parameter describing the interaction between each input unit and each output unit. This means every output unit interacts with every input unit. Convolutional networks, however, typically have sparse interactions (also referred to as sparse connectivity or sparse weights). This is accomplished by making the kernel smaller than the input. For example, when processing an image, the input image might have thousands or millions of pixels, but we can detect small, meaningful features such as edges with kernels that occupy only tens or hundreds of pixels. This means that we need to store fewer parameters, which both reduces the memory requirements of the model and improves its statistical eﬃciency. It also means that computing the output requires fewer operations. These improvements in eﬃciency are usually quite large. If there are m inputs and n outputs, then matrix multiplication requires m × n parameters and the algorithms used in practice have O(m × n ) runtime (per example). If we limit the number of connections each output may have to k, then the sparsely connected approach requires only k × n parameters and O(k × n ) runtime. For many practical applications, it is possible to obtain good performance on the machine learning task while keeping k several orders of magnitude smaller than m . For graphical demonstrations of sparse connectivity, see Fig. 9.2 and Fig. 9.3. In a deep convolutional network, units in the deeper layers may indirectly interact with a larger portion of the input, as shown in Fig. 9.4. This allows the network to eﬃciently describe complicated interactions between many variables by constructing such interactions from simple building blocks that each describe only sparse interactions. Parameter sharing refers to using the same parameter for more than one function in a model. In a traditional neural net, each element of the weight matrix is used exactly once when computing the output of a layer. It is multiplied by one element of the input and then never revisited. As a synonym for parameter sharing, one can say that a network has tied weights, because the value of the weight applied to one input is tied to the value of a weight applied elsewhere. In a convolutional neural net, each member of the kernel is used at every position of the input (except perhaps some of the boundary pixels, depending on the design decisions regarding the boundary). The parameter sharing used by the convolution operation means that rather than learning a separate set of parameters for every location, we learn 335

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Figure 9.2: Sparse connectivity, viewed from below: We highlight one input unit, x3 , and also highlight the output units in s that are aﬀected by this unit. (Top) When s is formed by convolution with a kernel of width 3, only three outputs are aﬀected by x. (Bottom) When s is formed by matrix multiplication, connectivity is no longer sparse, so all of the outputs are aﬀected by x3 .

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Figure 9.3: Sparse connectivity, viewed from above: We highlight one output unit, s3 , and also highlight the input units in x that aﬀect this unit. These units are known as the receptive ﬁeld of s 3 . (Top) When s is formed by convolution with a kernel of width 3, only three inputs aﬀect s3 . (Bottom) When s is formed by matrix multiplication, connectivity is no longer sparse, so all of the inputs aﬀect s3 .

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Figure 9.4: The receptive ﬁeld of the units in the deeper layers of a convolutional network is larger than the receptive ﬁeld of the units in the shallow layers. This eﬀect increases if the network includes architectural features like strided convolution (Fig. 9.12) or pooling (Sec. 9.3). This means that even though direct connections in a convolutional net are very sparse, units in the deeper layers can be indirectly connected to all or most of the input image. 337

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Figure 9.5: Parameter sharing: Black arrows indicate the connections that use a particular parameter in two diﬀerent models. (Top) The black arrows indicate uses of the central element of a 3-element kernel in a convolutional model. Due to parameter sharing, this single parameter is used at all input locations. (Bottom) The single black arrow indicates the use of the central element of the weight matrix in a fully connected model. This model has no parameter sharing so the parameter is used only once.

only one set. This does not aﬀect the runtime of forward propagation—it is still O(k × n)—but it does further reduce the storage requirements of the model to k parameters. Recall that k is usually several orders of magnitude less than m. Since m and n are usually roughly the same size, k is practically insigniﬁcant compared to m × n. Convolution is thus dramatically more eﬃcient than dense matrix multiplication in terms of the memory requirements and statistical eﬃciency. For a graphical depiction of how parameter sharing works, see Fig. 9.5. As an example of both of these ﬁrst two principles in action, Fig. 9.6 shows how sparse connectivity and parameter sharing can dramatically improve the eﬃciency of a linear function for detecting edges in an image. In the case of convolution, the particular form of parameter sharing causes the layer to have a property called equivariance to translation. To say a function is equivariant means that if the input changes, the output changes in the same way. Speciﬁcally, a function f(x) is equivariant to a function g if f (g(x)) = g(f(x)). In the case of convolution, if we let g be any function that translates the input, i.e., shifts it, then the convolution function is equivariant to g. For example, let I be a function giving image brightness at integer coordinates. Let g be a function mapping one image function to another image function, such that I = g(I ) is 338

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the image function with I (x, y) = I(x − 1, y). This shifts every pixel of I one unit to the right. If we apply this transformation to I , then apply convolution, the result will be the same as if we applied convolution to I , then applied the transformation g to the output. When processing time series data, this means that convolution produces a sort of timeline that shows when diﬀerent features appear in the input. If we move an event later in time in the input, the exact same representation of it will appear in the output, just later in time. Similarly with images, convolution creates a 2-D map of where certain features appear in the input. If we move the object in the input, its representation will move the same amount in the output. This is useful for when we know that some function of a small number of neighboring pixels is useful when applied to multiple input locations. For example, when processing images, it is useful to detect edges in the ﬁrst layer of a convolutional network. The same edges appear more or less everywhere in the image, so it is practical to share parameters across the entire image. In some cases, we may not wish to share parameters across the entire image. For example, if we are processing images that are cropped to be centered on an individual’s face, we probably want to extract diﬀerent features at diﬀerent locations—the part of the network processing the top of the face needs to look for eyebrows, while the part of the network processing the bottom of the face needs to look for a chin. Convolution is not naturally equivariant to some other transformations, such as changes in the scale or rotation of an image. Other mechanisms are necessary for handling these kinds of transformations. Finally, some kinds of data cannot be processed by neural networks deﬁned by matrix multiplication with a ﬁxed-shape matrix. Convolution enables processing of some of these kinds of data. We discuss this further in Sec. 9.7.

9.3

Pooling

A typical layer of a convolutional network consists of three stages (see Fig. 9.7). In the ﬁrst stage, the layer performs several convolutions in parallel to produce a set of linear activations. In the second stage, each linear activation is run through a nonlinear activation function, such as the rectiﬁed linear activation function. This stage is sometimes called the detector stage. In the third stage, we use a pooling function to modify the output of the layer further. A pooling function replaces the output of the net at a certain location with a summary statistic of the nearby outputs. For example, the max pooling (Zhou and Chellappa, 1988) operation reports the maximum output within a rectangular 339

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Figure 9.6: Eﬃciency of edge detection. The image on the right was formed by taking each pixel in the original image and subtracting the value of its neighboring pixel on the left. This shows the strength of all of the vertically oriented edges in the input image, which can be a useful operation for object detection. Both images are 280 pixels tall. The input image is 320 pixels wide while the output image is 319 pixels wide. This transformation can be described by a convolution kernel containing two elements, and requires 319 × 280 × 3 = 267, 960 ﬂoating point operations (two multiplications and one addition per output pixel) to compute using convolution. To describe the same transformation with a matrix multiplication would take 320 × 280 × 319 × 280, or over eight billion, entries in the matrix, making convolution four billion times more eﬃcient for representing this transformation. The straightforward matrix multiplication algorithm performs over sixteen billion ﬂoating point operations, making convolution roughly 60,000 times more eﬃcient computationally. Of course, most of the entries of the matrix would be zero. If we stored only the nonzero entries of the matrix, then both matrix multiplication and convolution would require the same number of ﬂoating point operations to compute. The matrix would still need to contain 2 × 319 × 280 = 178, 640 entries. Convolution is an extremely eﬃcient way of describing transformations that apply the same linear transformation of a small, local region across the entire input. (Photo credit: Paula Goodfellow)

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Complex layer terminology

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Figure 9.7: The components of a typical convolutional neural network layer. There are two commonly used sets of terminology for describing these layers. (Left) In this terminology, the convolutional net is viewed as a small number of relatively complex layers, with each layer having many “stages.” In this terminology, there is a one-to-one mapping between kernel tensors and network layers. In this book we generally use this terminology. (Right) In this terminology, the convolutional net is viewed as a larger number of simple layers; every step of processing is regarded as a layer in its own right. This means that not every “layer” has parameters.

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neighborhood. Other popular pooling functions include the average of a rectangular neighborhood, the L2 norm of a rectangular neighborhood, or a weighted average based on the distance from the central pixel. In all cases, pooling helps to make the representation become approximately invariant to small translations of the input. Invariance to translation means that if we translate the input by a small amount, the values of most of the pooled outputs do not change. See Fig. 9.8 for an example of how this works. Invariance to local translation can be a very useful property if we care more about whether some feature is present than exactly where it is. For example, when determining whether an image contains a face, we need not know the location of the eyes with pixel-perfect accuracy, we just need to know that there is an eye on the left side of the face and an eye on the right side of the face. In other contexts, it is more important to preserve the location of a feature. For example, if we want to ﬁnd a corner deﬁned by two edges meeting at a speciﬁc orientation, we need to preserve the location of the edges well enough to test whether they meet. The use of pooling can be viewed as adding an inﬁnitely strong prior that the function the layer learns must be invariant to small translations. When this assumption is correct, it can greatly improve the statistical eﬃciency of the network. Pooling over spatial regions produces invariance to translation, but if we pool over the outputs of separately parametrized convolutions, the features can learn which transformations to become invariant to (see Fig. 9.9). Because pooling summarizes the responses over a whole neighborhood, it is possible to use fewer pooling units than detector units, by reporting summary statistics for pooling regions spaced k pixels apart rather than 1 pixel apart. See Fig. 9.10 for an example. This improves the computational eﬃciency of the network because the next layer has roughly k times fewer inputs to process. When the number of parameters in the next layer is a function of its input size (such as when the next layer is fully connected and based on matrix multiplication) this reduction in the input size can also result in improved statistical eﬃciency and reduced memory requirements for storing the parameters. For many tasks, pooling is essential for handling inputs of varying size. For example, if we want to classify images of variable size, the input to the classiﬁcation layer must have a ﬁxed size. This is usually accomplished by varying the size of an oﬀset between pooling regions so that the classiﬁcation layer always receives the same number of summary statistics regardless of the input size. For example, the ﬁnal pooling layer of the network may be deﬁned to output four sets of summary statistics, one for each quadrant of an image, regardless of the image size. Some theoretical work gives guidance as to which kinds of pooling one should 342

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Figure 9.8: Max pooling introduces invariance. (Top) A view of the middle of the output of a convolutional layer. The bottom row shows outputs of the nonlinearity. The top row shows the outputs of max pooling, with a stride of one pixel between pooling regions and a pooling region width of three pixels. (Bottom) A view of the same network, after the input has been shifted to the right by one pixel. Every value in the bottom row has changed, but only half of the values in the top row have changed, because the max pooling units are only sensitive to the maximum value in the neighborhood, not its exact location.

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Large response in detector unit 1

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Figure 9.9: Example of learned invariances: A pooling unit that pools over multiple features that are learned with separate parameters can learn to be invariant to transformations of the input. Here we show how a set of three learned ﬁlters and a max pooling unit can learn to become invariant to rotation. All three ﬁlters are intended to detect a hand-written 5. Each ﬁlter attempts to match a slightly diﬀerent orientation of the 5. When a 5 appears in the input, the corresponding ﬁlter will match it and cause a large activation in a detector unit. The max pooling unit then has a large activation regardless of which pooling unit was activated. We show here how the network processes two diﬀerent inputs, resulting in two diﬀerent detector units being activated. The eﬀect on the pooling unit is roughly the same either way. This principle is leveraged by maxout networks (Goodfellow et al., 2013a) and other convolutional networks. Max pooling over spatial positions is naturally invariant to translation; this multi-channel approach is only necessary for learning other transformations.

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use in various situations (Boureau et al., 2010). It is also possible to dynamically pool features together, for example, by running a clustering algorithm on the locations of interesting features (Boureau et al., 2011). This approach yields a diﬀerent set of pooling regions for each image. Another approach is to learn a single pooling structure that is then applied to all images (Jia et al., 2012). Pooling can complicate some kinds of neural network architectures that use top-down information, such as Boltzmann machines and autoencoders. These issues will be discussed further when we present these types of networks in Part III. Pooling in convolutional Boltzmann machines is presented in Sec. 20.6. The inverse-like operations on pooling units needed in some diﬀerentiable networks will be covered in Sec. 20.10.6. Some examples of complete convolutional network architectures for classiﬁcation using convolution and pooling are shown in Fig. 9.11.

9.4

Convolution and Pooling as an Inﬁnitely Strong Prior

Recall the concept of a prior probability distribution from Sec. 5.2. This is a probability distribution over the parameters of a model that encodes our beliefs about what models are reasonable, before we have seen any data. Priors can be considered weak or strong depending on how concentrated the probability density in the prior is. A weak prior is a prior distribution with high entropy, such as a Gaussian distribution with high variance. Such a prior allows the data to move the parameters more or less freely. A strong prior has very low entropy, such as a Gaussian distribution with low variance. Such a prior plays a more active role in determining where the parameters end up. An inﬁnitely strong prior places zero probability on some parameters and says that these parameter values are completely forbidden, regardless of how much support the data gives to those values. We can imagine a convolutional net as being similar to a fully connected net, but with an inﬁnitely strong prior over its weights. This inﬁnitely strong prior says that the weights for one hidden unit must be identical to the weights of its neighbor, but shifted in space. The prior also says that the weights must be zero, except for in the small, spatially contiguous receptive ﬁeld assigned to that hidden unit. Overall, we can think of the use of convolution as introducing an inﬁnitely strong prior probability distribution over the parameters of a layer. This prior says that the function the layer should learn contains only local interactions and is 345

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Output of softmax: 1,000 class probabilities

Output of softmax: 1,000 class probabilities

Output of softmax: 1,000 class probabilities

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Figure 9.11: Examples of architectures for classiﬁcation with convolutional networks. The speciﬁc strides and depths used in this ﬁgure are not advisable for real use; they are designed to be very shallow in order to ﬁt onto the page. Real convolutional networks also often involve signiﬁcant amounts of branching, unlike the chain structures used here for simplicity. (Left) A convolutional network that processes a ﬁxed image size. After alternating between convolution and pooling for a few layers, the tensor for the convolutional feature map is reshaped to ﬂatten out the spatial dimensions. The rest of the network is an ordinary feedforward network classiﬁer, as described in Chapter 6. (Center) A convolutional network that processes a variable-sized image, but still maintains a fully connected section. This network uses a pooling operation with variably-sized pools but a ﬁxed number of pools, in order to provide a ﬁxed-size vector of 576 units to the fully connected portion of the network. (Right) A convolutional network that does not have any fully connected weight layer. Instead, the last convolutional layer outputs one feature map per class. The model presumably learns a map of how likely each class is to occur at each spatial location. Averaging a feature map down to a single value provides the argument to the softmax classiﬁer at the top. 346

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equivariant to translation. Likewise, the use of pooling is an inﬁnitely strong prior that each unit should be invariant to small translations. Of course, implementing a convolutional net as a fully connected net with an inﬁnitely strong prior would be extremely computationally wasteful. But thinking of a convolutional net as a fully connected net with an inﬁnitely strong prior can give us some insights into how convolutional nets work. One key insight is that convolution and pooling can cause underﬁtting. Like any prior, convolution and pooling are only useful when the assumptions made by the prior are reasonably accurate. If a task relies on preserving precise spatial information, then using pooling on all features can increase the training error. Some convolutional network architectures (Szegedy et al., 2014a) are designed to use pooling on some channels but not on other channels, in order to get both highly invariant features and features that will not underﬁt when the translation invariance prior is incorrect. When a task involves incorporating information from very distant locations in the input, then the prior imposed by convolution may be inappropriate. Another key insight from this view is that we should only compare convolutional models to other convolutional models in benchmarks of statistical learning performance. Models that do not use convolution would be able to learn even if we permuted all of the pixels in the image. For many image datasets, there are separate benchmarks for models that are permutation invariant and must discover the concept of topology via learning, and models that have the knowledge of spatial relationships hard-coded into them by their designer.

9.5

Variants of the Basic Convolution Function

When discussing convolution in the context of neural networks, we usually do not refer exactly to the standard discrete convolution operation as it is usually understood in the mathematical literature. The functions used in practice diﬀer slightly. Here we describe these diﬀerences in detail, and highlight some useful properties of the functions used in neural networks. First, when we refer to convolution in the context of neural networks, we usually actually mean an operation that consists of many applications of convolution in parallel. This is because convolution with a single kernel can only extract one kind of feature, albeit at many spatial locations. Usually we want each layer of our network to extract many kinds of features, at many locations. Additionally, the input is usually not just a grid of real values. Rather, it is a 347

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grid of vector-valued observations. For example, a color image has a red, green and blue intensity at each pixel. In a multilayer convolutional network, the input to the second layer is the output of the ﬁrst layer, which usually has the output of many diﬀerent convolutions at each position. When working with images, we usually think of the input and output of the convolution as being 3-D tensors, with one index into the diﬀerent channels and two indices into the spatial coordinates of each channel. Software implementations usually work in batch mode, so they will actually use 4-D tensors, with the fourth axis indexing diﬀerent examples in the batch, but we will omit the batch axis in our description here for simplicity. Because convolutional networks usually use multi-channel convolution, the linear operations they are based on are not guaranteed to be commutative, even if kernel-ﬂipping is used. These multi-channel operations are only commutative if each operation has the same number of output channels as input channels. Assume we have a 4-D kernel tensor K with element K i,j,k,l giving the connection strength between a unit in channel i of the output and a unit in channel j of the input, with an oﬀset of k rows and l columns between the output unit and the input unit. Assume our input consists of observed data V with element Vi,j,k giving the value of the input unit within channel i at row j and column k. Assume our output consists of Z with the same format as V. If Z is produced by convolving K across V without ﬂipping K, then (9.7) Zi,j,k = V l,j +m−1,k +n−1 Ki,l,m,n l,m,n

where the summation over l , m and n is over all values for which the tensor indexing operations inside the summation is valid. In linear algebra notation, we index into arrays using a 1 for the ﬁrst entry. This necessitates the − 1 in the above formula. Programming languages such as C and Python index starting from 0, rendering the above expression even simpler. We may want to skip over some positions of the kernel in order to reduce the computational cost (at the expense of not extracting our features as ﬁnely). We can think of this as downsampling the output of the full convolution function. If we want to sample only every s pixels in each direction in the output, then we can deﬁne a downsampled convolution function c such that (9.8) Zi,j,k = c(K, V, s)i,j,k = Vl,(j −1)×s+m,(k −1)×s+n Ki,l,m,n . l,m,n

We refer to s as the stride of this downsampled convolution. It is also possible to deﬁne a separate stride for each direction of motion. See Fig. 9.12 for an illustration. 348

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Figure 9.12: Convolution with a stride. In this example, we use a stride of two. (Top) Convolution with a stride length of two implemented in a single operation. (Bottom) Convolution with a stride greater than one pixel is mathematically equivalent to convolution with unit stride followed by downsampling. Obviously, the two-step approach involving downsampling is computationally wasteful, because it computes many values that are then discarded.

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One essential feature of any convolutional network implementation is the ability to implicitly zero-pad the input V in order to make it wider. Without this feature, the width of the representation shrinks by one pixel less than the kernel width at each layer. Zero padding the input allows us to control the kernel width and the size of the output independently. Without zero padding, we are forced to choose between shrinking the spatial extent of the network rapidly and using small kernels—both scenarios that signiﬁcantly limit the expressive power of the network. See Fig. 9.13 for an example. Three special cases of the zero-padding setting are worth mentioning. One is the extreme case in which no zero-padding is used whatsoever, and the convolution kernel is only allowed to visit positions where the entire kernel is contained entirely within the image. In MATLAB terminology, this is called valid convolution. In this case, all pixels in the output are a function of the same number of pixels in the input, so the behavior of an output pixel is somewhat more regular. However, the size of the output shrinks at each layer. If the input image has width m and the kernel has width k, the output will be of width m − k + 1. The rate of this shrinkage can be dramatic if the kernels used are large. Since the shrinkage is greater than 0, it limits the number of convolutional layers that can be included in the network. As layers are added, the spatial dimension of the network will eventually drop to 1 × 1, at which point additional layers cannot meaningfully be considered convolutional. Another special case of the zero-padding setting is when just enough zero-padding is added to keep the size of the output equal to the size of the input. MATLAB calls this same convolution. In this case, the network can contain as many convolutional layers as the available hardware can support, since the operation of convolution does not modify the architectural possibilities available to the next layer. However, the input pixels near the border inﬂuence fewer output pixels than the input pixels near the center. This can make the border pixels somewhat underrepresented in the model. This motivates the other extreme case, which MATLAB refers to as full convolution, in which enough zeroes are added for every pixel to be visited k times in each direction, resulting in an output image of width m + k − 1. In this case, the output pixels near the border are a function of fewer pixels than the output pixels near the center. This can make it diﬃcult to learn a single kernel that performs well at all positions in the convolutional feature map. Usually the optimal amount of zero padding (in terms of test set classiﬁcation accuracy) lies somewhere between “valid” and “same” convolution. In some cases, we do not actually want to use convolution, but rather locally connected layers (LeCun, 1986, 1989). In this case, the adjacency matrix in the graph of our MLP is the same, but every connection has its own weight, speciﬁed 350

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... ...

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Figure 9.13: The eﬀect of zero padding on network size: Consider a convolutional network with a kernel of width six at every layer. In this example, we do not use any pooling, so only the convolution operation itself shrinks the network size. (Top) In this convolutional network, we do not use any implicit zero padding. This causes the representation to shrink by ﬁve pixels at each layer. Starting from an input of sixteen pixels, we are only able to have three convolutional layers, and the last layer does not ever move the kernel, so arguably only two of the layers are truly convolutional. The rate of shrinking can be mitigated by using smaller kernels, but smaller kernels are less expressive and some shrinking is inevitable in this kind of architecture. (Bottom) By adding ﬁve implicit zeroes to each layer, we prevent the representation from shrinking with depth. This allows us to make an arbitrarily deep convolutional network.

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by a 6-D tensor W. The indices into W are respectively: i, the output channel, j, the output row, k, the output column, l, the input channel, m, the row oﬀset within the input, and n, the column oﬀset within the input. The linear part of a locally connected layer is then given by = [Vl,j +m−1,k +n−1 wi,j,k,l,m,n] . (9.9) Zi,j,k l,m,n

This is sometimes also called unshared convolution, because it is a similar operation to discrete convolution with a small kernel, but without sharing parameters across locations. Fig. 9.14 compares local connections, convolution, and full connections. Locally connected layers are useful when we know that each feature should be a function of a small part of space, but there is no reason to think that the same feature should occur across all of space. For example, if we want to tell if an image is a picture of a face, we only need to look for the mouth in the bottom half of the image. It can also be useful to make versions of convolution or locally connected layers in which the connectivity is further restricted, for example to constrain that each output channel i be a function of only a subset of the input channels l. A common way to do this is to make the ﬁrst m output channels connect to only the ﬁrst n input channels, the second m output channels connect to only the second n input channels, and so on. See Fig. 9.15 for an example. Modeling interactions between few channels allows the network to have fewer parameters in order to reduce memory consumption and increase statistical eﬃciency, and also reduces the amount of computation needed to perform forward and back-propagation. It accomplishes these goals without reducing the number of hidden units. Tiled convolution (Gregor and LeCun, 2010a; Le et al., 2010) oﬀers a compromise between a convolutional layer and a locally connected layer. Rather than learning a separate set of weights at every spatial location, we learn a set of kernels that we rotate through as we move through space. This means that immediately neighboring locations will have diﬀerent ﬁlters, like in a locally connected layer, but the memory requirements for storing the parameters will increase only by a factor of the size of this set of kernels, rather than the size of the entire output feature map. See Fig. 9.16 for a comparison of locally connected layers, tiled convolution, and standard convolution. To deﬁne tiled convolution algebraically, let k be a 6-D tensor, where two of the dimensions correspond to diﬀerent locations in the output map. Rather than having a separate index for each location in the output map, output locations cycle through a set of t diﬀerent choices of kernel stack in each direction. If t is equal to 352

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Figure 9.14: Comparison of local connections, convolution, and full connections. (Top) A locally connected layer with a patch size of two pixels. Each edge is labeled with a unique letter to show that each edge is associated with its own weight parameter. (Center) A convolutional layer with a kernel width of two pixels. This model has exactly the same connectivity as the locally connected layer. The diﬀerence lies not in which units interact with each other, but in how the parameters are shared. The locally connected layer has no parameter sharing. The convolutional layer uses the same two weights repeatedly across the entire input, as indicated by the repetition of the letters labeling each edge. (Bottom) A fully connected layer resembles a locally connected layer in the sense that each edge has its own parameter (there are too many to label explicitly with letters in this diagram). However, it does not have the restricted connectivity of the locally connected layer.

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Output Tensor

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Figure 9.15: A convolutional network with the ﬁrst two output channels connected to only the ﬁrst two input channels, and the second two output channels connected to only the second two input channels. 354

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Figure 9.16: A comparison of locally connected layers, tiled convolution, and standard convolution. All three have the same sets of connections between units, when the same size of kernel is used. This diagram illustrates the use of a kernel that is two pixels wide. The diﬀerences between the methods lies in how they share parameters. (Top) A locally connected layer has no sharing at all. We indicate that each connection has its own weight by labeling each connection with a unique letter. (Center) Tiled convolution has a set of t diﬀerent kernels. Here we illustrate the case of t = 2. One of these kernels has edges labeled “a” and “b,” while the other has edges labeled “c” and “d.” Each time we move one pixel to the right in the output, we move on to using a diﬀerent kernel. This means that, like the locally connected layer, neighboring units in the output have diﬀerent parameters. Unlike the locally connected layer, after we have gone through all t available kernels, we cycle back to the ﬁrst kernel. If two output units are separated by a multiple of t steps, then they share parameters. (Bottom) Traditional convolution is equivalent to tiled convolution with t = 1. There is only one kernel and it is applied everywhere, as indicated in the diagram by using the kernel with weights labeled “a” and “b” everywhere.

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the output width, this is the same as a locally connected layer. Zi,j,k = Vl,j +m−1,k +n−1Ki,l,m,n,j %t+1,k %t+1 ,

(9.10)

l,m,n

where % is the modulo operation, with t%t = 0, ( t + 1)%t = 1, etc. It is straightforward to generalize this equation to use a diﬀerent tiling range for each dimension. Both locally connected layers and tiled convolutional layers have an interesting interaction with max-pooling: the detector units of these layers are driven by diﬀerent ﬁlters. If these ﬁlters learn to detect diﬀerent transformed versions of the same underlying features, then the max-pooled units become invariant to the learned transformation (see Fig. 9.9). Convolutional layers are hard-coded to be invariant speciﬁcally to translation. Other operations besides convolution are usually necessary to implement a convolutional network. To perform learning, one must be able to compute the gradient with respect to the kernel, given the gradient with respect to the outputs. In some simple cases, this operation can be performed using the convolution operation, but many cases of interest, including the case of stride greater than 1, do not have this property. Recall that convolution is a linear operation and can thus be described as a matrix multiplication (if we ﬁrst reshape the input tensor into a ﬂat vector). The matrix involved is a function of the convolution kernel. The matrix is sparse and each element of the kernel is copied to several elements of the matrix. This view helps us to derive some of the other operations needed to implement a convolutional network. Multiplication by the transpose of the matrix deﬁned by convolution is one such operation. This is the operation needed to back-propagate error derivatives through a convolutional layer, so it is needed to train convolutional networks that have more than one hidden layer. This same operation is also needed if we wish to reconstruct the visible units from the hidden units (Simard et al., 1992). Reconstructing the visible units is an operation commonly used in the models described in Part III of this book, such as autoencoders, RBMs, and sparse coding. Transpose convolution is necessary to construct convolutional versions of those models. Like the kernel gradient operation, this input gradient operation can be implemented using a convolution in some cases, but in the general case requires a third operation to be implemented. Care must be taken to coordinate this transpose operation with the forward propagation. The size of the output that the transpose operation should return depends on the zero padding policy and stride of 356

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the forward propagation operation, as well as the size of the forward propagation’s output map. In some cases, multiple sizes of input to forward propagation can result in the same size of output map, so the transpose operation must be explicitly told what the size of the original input was. These three operations—convolution, backprop from output to weights, and backprop from output to inputs—are suﬃcient to compute all of the gradients needed to train any depth of feedforward convolutional network, as well as to train convolutional networks with reconstruction functions based on the transpose of convolution. See Goodfellow (2010) for a full derivation of the equations in the fully general multi-dimensional, multi-example case. To give a sense of how these equations work, we present the two dimensional, single example version here. Suppose we want to train a convolutional network that incorporates strided convolution of kernel stack K applied to multi-channel image V with stride s as deﬁned by c(K, V, s) as in Eq. 9.8. Suppose we want to minimize some loss function J(V, K ). During forward propagation, we will need to use c itself to output Z, which is then propagated through the rest of the network and used to compute the cost function J. During back-propagation, we will receive a tensor G such that Gi,j,k = ∂Z ∂ J (V, K). i,j,k To train the network, we need to compute the derivatives with respect to the weights in the kernel. To do so, we can use a function g (G, V, s)i,j,k,l =

∂ ∂Ki,j,k,l

J (V, K) =

m,n

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(9.11)

If this layer is not the bottom layer of the network, we will need to compute the gradient with respect to V in order to back-propagate the error farther down. To do so, we can use a function h(K, G, s) i,j,k = =

∂ J (V, K) ∂Vi,j,k

(9.12)

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Kq,i,m,p Gq,l,n .

(9.13)

q

Autoencoder networks, described in Chapter 14, are feedforward networks trained to copy their input to their output. A simple example is the PCA algorithm, that copies its input x to an approximate reconstruction r using the function W W x. It is common for more general autoencoders to use multiplication by the transpose of the weight matrix just as PCA does. To make such models 357

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convolutional, we can use the function h to perform the transpose of the convolution operation. Suppose we have hidden units H in the same format as Z and we deﬁne a reconstruction R = h(K, H, s). (9.14) In order to train the autoencoder, we will receive the gradient with respect to R as a tensor E . To train the decoder, we need to obtain the gradient with respect to K. This is given by g(H, E, s). To train the encoder, we need to obtain the gradient with respect to H . This is given by c(K, E, s). It is also possible to diﬀerentiate through g using c and h, but these operations are not needed for the back-propagation algorithm on any standard network architectures. Generally, we do not use only a linear operation in order to transform from the inputs to the outputs in a convolutional layer. We generally also add some bias term to each output before applying the nonlinearity. This raises the question of how to share parameters among the biases. For locally connected layers it is natural to give each unit its own bias, and for tiled convolution, it is natural to share the biases with the same tiling pattern as the kernels. For convolutional layers, it is typical to have one bias per channel of the output and share it across all locations within each convolution map. However, if the input is of known, ﬁxed size, it is also possible to learn a separate bias at each location of the output map. Separating the biases may slightly reduce the statistical eﬃciency of the model, but also allows the model to correct for diﬀerences in the image statistics at diﬀerent locations. For example, when using implicit zero padding, detector units at the edge of the image receive less total input and may need larger biases.

9.6

Structured Outputs

Convolutional networks can be used to output a high-dimensional, structured object, rather than just predicting a class label for a classiﬁcation task or a real value for a regression task. Typically this object is just a tensor, emitted by a standard convolutional layer. For example, the model might emit a tensor S, where Si,j,k is the probability that pixel (j, k) of the input to the network belongs to class i. This allows the model to label every pixel in an image and draw precise masks that follow the outlines of individual objects. One issue that often comes up is that the output plane can be smaller than the input plane, as shown in Fig. 9.13. In the kinds of architectures typically used for classiﬁcation of a single object in an image, the greatest reduction in the spatial dimensions of the network comes from using pooling layers with large stride. In 358

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Figure 9.17: An example of a recurrent convolutional network for pixel labeling. The input is an image tensor X, with axes corresponding to image rows, image columns, and channels (red, green, blue). The goal is to output a tensor of labels Yˆ , with a probability distribution over labels for each pixel. This tensor has axes corresponding to image rows, image columns, and the diﬀerent classes. Rather than outputting Yˆ in a single shot, the recurrent network iteratively reﬁnes its estimate Yˆ by using a previous estimate of Yˆ as input for creating a new estimate. The same parameters are used for each updated estimate, and the estimate can be reﬁned as many times as we wish. The tensor of convolution kernels U is used on each step to compute the hidden representation given the input image. The kernel tensor V is used to produce an estimate of the labels given the hidden values. On all but the ﬁrst step, the kernels W are convolved over Yˆ to provide input to the hidden layer. On the ﬁrst time step, this term is replaced by zero. Because the same parameters are used on each step, this is an example of a recurrent network, as described in Chapter 10.

order to produce an output map of similar size as the input, one can avoid pooling altogether (Jain et al., 2007). Another strategy is to simply emit a lower-resolution grid of labels (Pinheiro and Collobert, 2014, 2015). Finally, in principle, one could use a pooling operator with unit stride. One strategy for pixel-wise labeling of images is to produce an initial guess of the image labels, then reﬁne this initial guess using the interactions between neighboring pixels. Repeating this reﬁnement step several times corresponds to using the same convolutions at each stage, sharing weights between the last layers of the deep net (Jain et al., 2007). This makes the sequence of computations performed by the successive convolutional layers with weights shared across layers a particular kind of recurrent network (Pinheiro and Collobert, 2014, 2015). Fig. 9.17 shows the architecture of such a recurrent convolutional network. Once a prediction for each pixel is made, various methods can be used to further process these predictions in order to obtain a segmentation of the image into regions (Briggman et al., 2009; Turaga et al., 2010; Farabet et al., 2013). 359

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The general idea is to assume that large groups of contiguous pixels tend to be associated with the same label. Graphical models can describe the probabilistic relationships between neighboring pixels. Alternatively, the convolutional network can be trained to maximize an approximation of the graphical model training objective (Ning et al., 2005; Thompson et al., 2014).

9.7

Data Types

The data used with a convolutional network usually consists of several channels, each channel being the observation of a diﬀerent quantity at some point in space or time. See Table 9.1 for examples of data types with diﬀerent dimensionalities and number of channels. For an example of convolutional networks applied to video, see Chen et al. (2010). So far we have discussed only the case where every example in the train and test data has the same spatial dimensions. One advantage to convolutional networks is that they can also process inputs with varying spatial extents. These kinds of input simply cannot be represented by traditional, matrix multiplication-based neural networks. This provides a compelling reason to use convolutional networks even when computational cost and overﬁtting are not signiﬁcant issues. For example, consider a collection of images, where each image has a diﬀerent width and height. It is unclear how to model such inputs with a weight matrix of ﬁxed size. Convolution is straightforward to apply; the kernel is simply applied a diﬀerent number of times depending on the size of the input, and the output of the convolution operation scales accordingly. Convolution may be viewed as matrix multiplication; the same convolution kernel induces a diﬀerent size of doubly block circulant matrix for each size of input. Sometimes the output of the network is allowed to have variable size as well as the input, for example if we want to assign a class label to each pixel of the input. In this case, no further design work is necessary. In other cases, the network must produce some ﬁxed-size output, for example if we want to assign a single class label to the entire image. In this case we must make some additional design steps, like inserting a pooling layer whose pooling regions scale in size proportional to the size of the input, in order to maintain a ﬁxed number of pooled outputs. Some examples of this kind of strategy are shown in Fig. 9.11. Note that the use of convolution for processing variable sized inputs only makes sense for inputs that have variable size because they contain varying amounts 360

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1-D

2-D

3-D

Single channel Audio waveform: The axis we convolve over corresponds to time. We discretize time and measure the amplitude of the waveform once per time step.

Audio data that has been preprocessed with a Fourier transform: We can transform the audio waveform into a 2D tensor with different rows corresponding to different frequencies and diﬀerent columns corresponding to diﬀerent points in time. Using convolution in the time makes the model equivariant to shifts in time. Using convolution across the frequency axis makes the model equivariant to frequency, so that the same melody played in a different octave produces the same representation but at a diﬀerent height in the network’s output. Volumetric data: A common source of this kind of data is medical imaging technology, such as CT scans.

Multi-channel Skeleton animation data: Animations of 3-D computer-rendered characters are generated by altering the pose of a “skeleton” over time. At each point in time, the pose of the character is described by a speciﬁcation of the angles of each of the joints in the character’s skeleton. Each channel in the data we feed to the convolutional model represents the angle about one axis of one joint. Color image data: One channel contains the red pixels, one the green pixels, and one the blue pixels. The convolution kernel moves over both the horizontal and vertical axes of the image, conferring translation equivariance in both directions.

Color video data: One axis corresponds to time, one to the height of the video frame, and one to the width of the video frame.

Table 9.1: Examples of diﬀerent formats of data that can be used with convolutional networks.

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of observation of the same kind of thing—diﬀerent lengths of recordings over time, diﬀerent widths of observations over space, etc. Convolution does not make sense if the input has variable size because it can optionally include diﬀerent kinds of observations. For example, if we are processing college applications, and our features consist of both grades and standardized test scores, but not every applicant took the standardized test, then it does not make sense to convolve the same weights over both the features corresponding to the grades and the features corresponding to the test scores.

9.8

Eﬃcient Convolution Algorithms

Modern convolutional network applications often involve networks containing more than one million units. Powerful implementations exploiting parallel computation resources, as discussed in Sec. 12.1, are essential. However, in many cases it is also possible to speed up convolution by selecting an appropriate convolution algorithm. Convolution is equivalent to converting both the input and the kernel to the frequency domain using a Fourier transform, performing point-wise multiplication of the two signals, and converting back to the time domain using an inverse Fourier transform. For some problem sizes, this can be faster than the naive implementation of discrete convolution. When a d-dimensional kernel can be expressed as the outer product of d vectors, one vector per dimension, the kernel is called separable. When the kernel is separable, naive convolution is ineﬃcient. It is equivalent to compose d onedimensional convolutions with each of these vectors. The composed approach is signiﬁcantly faster than performing one d-dimensional convolution with their outer product. The kernel also takes fewer parameters to represent as vectors. If the kernel is w elements wide in each dimension, then naive multidimensional convolution requires O (w d ) runtime and parameter storage space, while separable convolution requires O(w × d) runtime and parameter storage space. Of course, not every convolution can be represented in this way. Devising faster ways of performing convolution or approximate convolution without harming the accuracy of the model is an active area of research. Even techniques that improve the eﬃciency of only forward propagation are useful because in the commercial setting, it is typical to devote more resources to deployment of a network than to its training.

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9.9

Random or Unsupervised Features

Typically, the most expensive part of convolutional network training is learning the features. The output layer is usually relatively inexpensive due to the small number of features provided as input to this layer after passing through several layers of pooling. When performing supervised training with gradient descent, every gradient step requires a complete run of forward propagation and backward propagation through the entire network. One way to reduce the cost of convolutional network training is to use features that are not trained in a supervised fashion. There are three basic strategies for obtaining convolution kernels without supervised training. One is to simply initialize them randomly. Another is to design them by hand, for example by setting each kernel to detect edges at a certain orientation or scale. Finally, one can learn the kernels with an unsupervised criterion. For example, Coates et al. (2011) apply k-means clustering to small image patches, then use each learned centroid as a convolution kernel. Part III describes many more unsupervised learning approaches. Learning the features with an unsupervised criterion allows them to be determined separately from the classiﬁer layer at the top of the architecture. One can then extract the features for the entire training set just once, essentially constructing a new training set for the last layer. Learning the last layer is then typically a convex optimization problem, assuming the last layer is something like logistic regression or an SVM. Random ﬁlters often work surprisingly well in convolutional networks (Jarrett et al., 2009; Saxe et al., 2011; Pinto et al., 2011; Cox and Pinto, 2011). Saxe et al. (2011) showed that layers consisting of convolution following by pooling naturally become frequency selective and translation invariant when assigned random weights. They argue that this provides an inexpensive way to choose the architecture of a convolutional network: ﬁrst evaluate the performance of several convolutional network architectures by training only the last layer, then take the best of these architectures and train the entire architecture using a more expensive approach. An intermediate approach is to learn the features, but using methods that do not require full forward and back-propagation at every gradient step. As with multilayer perceptrons, we use greedy layer-wise pretraining, to train the ﬁrst layer in isolation, then extract all features from the ﬁrst layer only once, then train the second layer in isolation given those features, and so on. Chapter 8 has described how to perform supervised greedy layer-wise pretraining, and Part III extends this to greedy layer-wise pretraining using an unsupervised criterion at each layer. The canonical example of greedy layer-wise pretraining of a convolutional model is the convolutional deep belief network (Lee et al., 2009). Convolutional networks oﬀer 363

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us the opportunity to take the pretraining strategy one step further than is possible with multilayer perceptrons. Instead of training an entire convolutional layer at a time, we can train a model of a small patch, as Coates et al. (2011) do with k-means. We can then use the parameters from this patch-based model to deﬁne the kernels of a convolutional layer. This means that it is possible to use unsupervised learning to train a convolutional network without ever using convolution during the training process. Using this approach, we can train very large models and incur a high computational cost only at inference time (Ranzato et al., 2007b; Jarrett et al., 2009; Kavukcuoglu et al., 2010; Coates et al., 2013). This approach was popular from roughly 2007–2013, when labeled datasets were small and computational power was more limited. Today, most convolutional networks are trained in a purely supervised fashion, using full forward and back-propagation through the entire network on each training iteration. As with other approaches to unsupervised pretraining, it remains diﬃcult to tease apart the cause of some of the beneﬁts seen with this approach. Unsupervised pretraining may oﬀer some regularization relative to supervised training, or it may simply allow us to train much larger architectures due to the reduced computational cost of the learning rule.

9.10

The Neuroscientiﬁc Basis for Convolutional Networks

Convolutional networks are perhaps the greatest success story of biologically inspired artiﬁcial intelligence. Though convolutional networks have been guided by many other ﬁelds, some of the key design principles of neural networks were drawn from neuroscience. The history of convolutional networks begins with neuroscientiﬁc experiments long before the relevant computational models were developed. Neurophysiologists David Hubel and Torsten Wiesel collaborated for several years to determine many of the most basic facts about how the mammalian vision system works (Hubel and Wiesel, 1959, 1962, 1968). Their accomplishments were eventually recognized with a Nobel prize. Their ﬁndings that have had the greatest inﬂuence on contemporary deep learning models were based on recording the activity of individual neurons in cats. They observed how neurons in the cat’s brain responded to images projected in precise locations on a screen in front of the cat. Their great discovery was that neurons in the early visual system responded most strongly to very speciﬁc patterns of light, such as precisely oriented bars, but responded hardly at all to other patterns. 364

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Their work helped to characterize many aspects of brain function that are beyond the scope of this book. From the point of view of deep learning, we can focus on a simpliﬁed, cartoon view of brain function. In this simpliﬁed view, we focus on a part of the brain called V1, also known as the primary visual cortex. V1 is the ﬁrst area of the brain that begins to perform signiﬁcantly advanced processing of visual input. In this cartoon view, images are formed by light arriving in the eye and stimulating the retina, the light-sensitive tissue in the back of the eye. The neurons in the retina perform some simple preprocessing of the image but do not substantially alter the way it is represented. The image then passes through the optic nerve and a brain region called the lateral geniculate nucleus. The main role, as far as we are concerned here, of both of these anatomical regions is primarily just to carry the signal from the eye to V1, which is located at the back of the head. A convolutional network layer is designed to capture three properties of V1: 1. V1 is arranged in a spatial map. It actually has a two-dimensional structure mirroring the structure of the image in the retina. For example, light arriving at the lower half of the retina aﬀects only the corresponding half of V1. Convolutional networks capture this property by having their features deﬁned in terms of two dimensional maps. 2. V1 contains many simple cells. A simple cell’s activity can to some extent be characterized by a linear function of the image in a small, spatially localized receptive ﬁeld. The detector units of a convolutional network are designed to emulate these properties of simple cells. 3. V1 also contains many complex cells. These cells respond to features that are similar to those detected by simple cells, but complex cells are invariant to small shifts in the position of the feature. This inspires the pooling units of convolutional networks. Complex cells are also invariant to some changes in lighting that cannot be captured simply by pooling over spatial locations. These invariances have inspired some of the cross-channel pooling strategies in convolutional networks, such as maxout units (Goodfellow et al., 2013a). Though we know the most about V1, it is generally believed that the same basic principles apply to other areas of the visual system. In our cartoon view of the visual system, the basic strategy of detection followed by pooling is repeatedly applied as we move deeper into the brain. As we pass through multiple anatomical layers of the brain, we eventually ﬁnd cells that respond to some speciﬁc concept and are invariant to many transformations of the input. These cells have been 365

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nicknamed “grandmother cells”—the idea is that a person could have a neuron that activates when seeing an image of their grandmother, regardless of whether she appears in the left or right side of the image, whether the image is a close-up of her face or zoomed out shot of her entire body, whether she is brightly lit, or in shadow, etc. These grandmother cells have been shown to actually exist in the human brain, in a region called the medial temporal lobe (Quiroga et al., 2005). Researchers tested whether individual neurons would respond to photos of famous individuals. They found what has come to be called the “Halle Berry neuron”: an individual neuron that is activated by the concept of Halle Berry. This neuron ﬁres when a person sees a photo of Halle Berry, a drawing of Halle Berry, or even text containing the words “Halle Berry.” Of course, this has nothing to do with Halle Berry herself; other neurons responded to the presence of Bill Clinton, Jennifer Aniston, etc. These medial temporal lobe neurons are somewhat more general than modern convolutional networks, which would not automatically generalize to identifying a person or object when reading its name. The closest analog to a convolutional network’s last layer of features is a brain area called the inferotemporal cortex (IT). When viewing an object, information ﬂows from the retina, through the LGN, to V1, then onward to V2, then V4, then IT. This happens within the ﬁrst 100ms of glimpsing an object. If a person is allowed to continue looking at the object for more time, then information will begin to ﬂow backwards as the brain uses top-down feedback to update the activations in the lower level brain areas. However, if we interrupt the person’s gaze, and observe only the ﬁring rates that result from the ﬁrst 100ms of mostly feedforward activation, then IT proves to be very similar to a convolutional network. Convolutional networks can predict IT ﬁring rates, and also perform very similarly to (time limited) humans on object recognition tasks (DiCarlo, 2013). That being said, there are many diﬀerences between convolutional networks and the mammalian vision system. Some of these diﬀerences are well known to computational neuroscientists, but outside the scope of this book. Some of these diﬀerences are not yet known, because many basic questions about how the mammalian vision system works remain unanswered. As a brief list: • The human eye is mostly very low resolution, except for a tiny patch called the fovea. The fovea only observes an area about the size of a thumbnail held at arms length. Though we feel as if we can see an entire scene in high resolution, this is an illusion created by the subconscious part of our brain, as it stitches together several glimpses of small areas. Most convolutional networks actually receive large full resolution photographs as input. The human brain makes 366

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several eye movements called saccades to glimpse the most visually salient or task-relevant parts of a scene. Incorporating similar attention mechanisms into deep learning models is an active research direction. In the context of deep learning, attention mechanisms have been most successful for natural language processing, as described in Sec. 12.4.5.1. Several visual models with foveation mechanisms have been developed but so far have not become the dominant approach (Larochelle and Hinton, 2010; Denil et al., 2012). • The human visual system is integrated with many other senses, such as hearing, and factors like our moods and thoughts. Convolutional networks so far are purely visual. • The human visual system does much more than just recognize objects. It is able to understand entire scenes including many objects and relationships between objects, and processes rich 3-D geometric information needed for our bodies to interface with the world. Convolutional networks have been applied to some of these problems but these applications are in their infancy. • Even simple brain areas like V1 are heavily impacted by feedback from higher levels. Feedback has been explored extensively in neural network models but has not yet been shown to oﬀer a compelling improvement. • While feedforward IT ﬁring rates capture much of the same information as convolutional network features, it is not clear how similar the intermediate computations are. The brain probably uses very diﬀerent activation and pooling functions. An individual neuron’s activation probably is not wellcharacterized by a single linear ﬁlter response. A recent model of V1 involves multiple quadratic ﬁlters for each neuron (Rust et al., 2005). Indeed our cartoon picture of “simple cells” and “complex cells” might create a nonexistent distinction; simple cells and complex cells might both be the same kind of cell but with their “parameters” enabling a continuum of behaviors ranging from what we call “simple” to what we call “complex.” It is also worth mentioning that neuroscience has told us relatively little about how to train convolutional networks. Model structures with parameter sharing across multiple spatial locations date back to early connectionist models of vision (Marr and Poggio, 1976), but these models did not use the modern back-propagation algorithm and gradient descent. For example, the Neocognitron (Fukushima, 1980) incorporated most of the model architecture design elements of the modern convolutional network but relied on a layer-wise unsupervised clustering algorithm. 367

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Lang and Hinton (1988) introduced the use of back-propagation to train timedelay neural networks (TDNNs). To use contemporary terminology, TDNNs are one-dimensional convolutional networks applied to time series. Back-propagation applied to these models was not inspired by any neuroscientiﬁc observation and is considered by some to be biologically implausible. Following the success of back-propagation-based training of TDNNs, (LeCun et al., 1989) developed the modern convolutional network by applying the same training algorithm to 2-D convolution applied to images. So far we have described how simple cells are roughly linear and selective for certain features, complex cells are more nonlinear and become invariant to some transformations of these simple cell features, and stacks of layers that alternate between selectivity and invariance can yield grandmother cells for very speciﬁc phenomena. We have not yet described precisely what these individual cells detect. In a deep, nonlinear network, it can be diﬃcult to understand the function of individual cells. Simple cells in the ﬁrst layer are easier to analyze, because their responses are driven by a linear function. In an artiﬁcial neural network, we can just display an image of the convolution kernel to see what the corresponding channel of a convolutional layer responds to. In a biological neural network, we do not have access to the weights themselves. Instead, we put an electrode in the neuron itself, display several samples of white noise images in front of the animal’s retina, and record how each of these samples causes the neuron to activate. We can then ﬁt a linear model to these responses in order to obtain an approximation of the neuron’s weights. This approach is known as reverse correlation (Ringach and Shapley, 2004). Reverse correlation shows us that most V1 cells have weights that are described by Gabor functions. The Gabor function describes the weight at a 2-D point in the image. We can think of an image as being a function of 2-D coordinates, I (x, y). Likewise, we can think of a simple cell as sampling the image at a set of locations, deﬁned by a set of x coordinates X and a set of y coordinates, Y, and applying weights that are also a function of the location, w(x, y). From this point of view, the response of a simple cell to an image is given by s(I ) = w (x, y )I (x, y ). (9.15) x∈X y∈Y

Speciﬁcally, w (x, y ) takes the form of a Gabor function: w (x, y ; α, βx , βy , f, φ, x 0, y 0, τ ) = α exp −β x x 2 − β yy 2 cos(fx + φ), where

x = (x − x0 ) cos(τ ) + (y − y0) sin(τ ) 368

(9.16) (9.17)

CHAPTER 9. CONVOLUTIONAL NETWORKS

and y = −(x − x 0) sin(τ ) + (y − y0) cos(τ ).

(9.18)

Here, α, β x, βy , f, φ, x 0, y0, and τ are parameters that control the properties of the Gabor function. Fig. 9.18 shows some examples of Gabor functions with diﬀerent settings of these parameters. The parameters x0, y0 , and τ deﬁne a coordinate system. We translate and rotate x and y to form x and y . Speciﬁcally, the simple cell will respond to image features centered at the point (x 0, y 0), and it will respond to changes in brightness as we move along a line rotated τ radians from the horizontal. Viewed as a function of x and y , the function w then responds to changes in brightness as we move along the x axis. It has two important factors: one is a Gaussian function and the other is a cosine function. The Gaussian factor α exp −βx x2 − β y y2 can be seen as a gating term that ensures the simple cell will only respond to values near where x and y are both zero, in other words, near the center of the cell’s receptive ﬁeld. The scaling factor α adjusts the total magnitude of the simple cell’s response, while βx and β y control how quickly its receptive ﬁeld falls oﬀ. The cosine factor cos (fx + φ) controls how the simple cell responds to changing brightness along the x axis. The parameter f controls the frequency of the cosine and φ controls its phase oﬀset. Altogether, this cartoon view of simple cells means that a simple cell responds to a speciﬁc spatial frequency of brightness in a speciﬁc direction at a speciﬁc location. Simple cells are most excited when the wave of brightness in the image has the same phase as the weights. This occurs when the image is bright where the weights are positive and dark where the weights are negative. Simple cells are most inhibited when the wave of brightness is fully out of phase with the weights—when the image is dark where the weights are positive and bright where the weights are negative. 2 The cartoon view of a complex cell is that it computes the L norm of the 2-D vector containing two simple cells’ responses: c(I) = s0(I )2 + s1(I ) 2. An important special case occurs when s 1 has all of the same parameters as s0 except for φ, and φ is set such that s1 is one quarter cycle out of phase with s0 . In this case, s0 and s1 form a quadrature pair. A complex cell deﬁned in this way responds when the Gaussian reweighted image I(x, y) exp(−βx x2 − β yy 2) contains a high amplitude sinusoidal wave with frequency f in direction τ near (x 0 , y0), regardless of the phase oﬀset of this wave. In other words, the complex cell is invariant to small translations of the image in direction τ , or to negating the

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Figure 9.18: Gabor functions with a variety of parameter settings. White indicates large positive weight, black indicates large negative weight, and the background gray corresponds to zero weight. (Left) Gabor functions with diﬀerent values of the parameters that control the coordinate system: x 0, y0 , and τ. Each Gabor function in this grid is assigned a value of x0 and y 0 proportional to its position in its grid, and τ is chosen so that each Gabor ﬁlter is sensitive to the direction radiating out from the center of the grid. For the other two plots, x0 , y0 , and τ are ﬁxed to zero. (Center) Gabor functions with diﬀerent Gaussian scale parameters βx and βy . Gabor functions are arranged in increasing width (decreasing β x) as we move left to right through the grid, and increasing height (decreasing βy ) as we move top to bottom. For the other two plots, the β values are ﬁxed to 1.5× the image width. (Right) Gabor functions with diﬀerent sinusoid parameters f and φ. As we move top to bottom, f increases, and as we move left to right, φ increases. For the other two plots, φ is ﬁxed to 0 and f is ﬁxed to 5× the image width.

image (replacing black with white and vice versa). Some of the most striking correspondences between neuroscience and machine learning come from visually comparing the features learned by machine learning models with those employed by V1. Olshausen and Field (1996) showed that a simple unsupervised learning algorithm, sparse coding, learns features with receptive ﬁelds similar to those of simple cells. Since then, we have found that an extremely wide variety of statistical learning algorithms learn features with Gabor-like functions when applied to natural images. This includes most deep learning algorithms, which learn these features in their ﬁrst layer. Fig. 9.19 shows some examples. Because so many diﬀerent learning algorithms learn edge detectors, it is diﬃcult to conclude that any speciﬁc learning algorithm is the “right” model of the brain just based on the features that it learns (though it can certainly be a bad sign if an algorithm does not learn some sort of edge detector when applied to natural images). These features are an important part of the statistical structure of natural images and can be recovered by many diﬀerent approaches to statistical modeling. See Hyvärinen et al. (2009) for a review of the ﬁeld of natural image statistics.

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Figure 9.19: Many machine learning algorithms learn features that detect edges or speciﬁc colors of edges when applied to natural images. These feature detectors are reminiscent of the Gabor functions known to be present in primary visual cortex. (Left) Weights learned by an unsupervised learning algorithm (spike and slab sparse coding) applied to small image patches. (Right) Convolution kernels learned by the ﬁrst layer of a fully supervised convolutional maxout network. Neighboring pairs of ﬁlters drive the same maxout unit.

9.11

Convolutional Networks and the History of Deep Learning

Convolutional networks have played an important role in the history of deep learning. They are a key example of a successful application of insights obtained by studying the brain to machine learning applications. They were also some of the ﬁrst deep models to perform well, long before arbitrary deep models were considered viable. Convolutional networks were also some of the ﬁrst neural networks to solve important commercial applications and remain at the forefront of commercial applications of deep learning today. For example, in the 1990s, the neural network research group at AT&T developed a convolutional network for reading checks (LeCun et al., 1998b). By the end of the 1990s, this system deployed by NEC was reading over 10% of all the checks in the US. Later, several OCR and handwriting recognition systems based on convolutional nets were deployed by Microsoft (Simard et al., 2003). See Chapter 12 for more details on such applications and more modern applications of convolutional networks. See LeCun et al. (2010) for a more in-depth history of convolutional networks up to 2010. Convolutional networks were also used to win many contests. The current intensity of commercial interest in deep learning began when Krizhevsky et al. (2012) won the ImageNet object recognition challenge, but convolutional networks 371

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had been used to win other machine learning and computer vision contests with less impact for years earlier. Convolutional nets were some of the ﬁrst working deep networks trained with back-propagation. It is not entirely clear why convolutional networks succeeded when general back-propagation networks were considered to have failed. It may simply be that convolutional networks were more computationally eﬃcient than fully connected networks, so it was easier to run multiple experiments with them and tune their implementation and hyperparameters. Larger networks also seem to be easier to train. With modern hardware, large fully connected networks appear to perform reasonably on many tasks, even when using datasets that were available and activation functions that were popular during the times when fully connected networks were believed not to work well. It may be that the primary barriers to the success of neural networks were psychological (practitioners did not expect neural networks to work, so they did not make a serious eﬀort to use neural networks). Whatever the case, it is fortunate that convolutional networks performed well decades ago. In many ways, they carried the torch for the rest of deep learning and paved the way to the acceptance of neural networks in general. Convolutional networks provide a way to specialize neural networks to work with data that has a clear grid-structured topology and to scale such models to very large size. This approach has been the most successful on a two-dimensional, image topology. To process one-dimensional, sequential data, we turn next to another powerful specialization of the neural networks framework: recurrent neural networks.

372

Chapter 10

Sequence Modeling: Recurrent and Recursive Nets Recurrent neural networks or RNNs (Rumelhart et al., 1986a) are a family of neural networks for processing sequential data. Much as a convolutional network is a neural network that is specialized for processing a grid of values X such as an image, a recurrent neural network is a neural network that is specialized for processing a sequence of values x(1) , . . . , x(τ ) . Just as convolutional networks can readily scale to images with large width and height, and some convolutional networks can process images of variable size, recurrent networks can scale to much longer sequences than would be practical for networks without sequence-based specialization. Most recurrent networks can also process sequences of variable length. To go from multi-layer networks to recurrent networks, we need to take advantage of one of the early ideas found in machine learning and statistical models of the 1980s: sharing parameters across diﬀerent parts of a model. Parameter sharing makes it possible to extend and apply the model to examples of diﬀerent forms (diﬀerent lengths, here) and generalize across them. If we had separate parameters for each value of the time index, we could not generalize to sequence lengths not seen during training, nor share statistical strength across diﬀerent sequence lengths and across diﬀerent positions in time. Such sharing is particularly important when a speciﬁc piece of information can occur at multiple positions within the sequence. For example, consider the two sentences “I went to Nepal in 2009” and “In 2009, I went to Nepal.” If we ask a machine learning model to read each sentence and extract the year in which the narrator went to Nepal, we would like it to recognize the year 2009 as the relevant piece of information, whether it appears in the sixth 373

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word or the second word of the sentence. Suppose that we trained a feedforward network that processes sentences of ﬁxed length. A traditional fully connected feedforward network would have separate parameters for each input feature, so it would need to learn all of the rules of the language separately at each position in the sentence. By comparison, a recurrent neural network shares the same weights across several time steps. A related idea is the use of convolution across a 1-D temporal sequence. This convolutional approach is the basis for time-delay neural networks (Lang and Hinton, 1988; Waibel et al., 1989; Lang et al., 1990). The convolution operation allows a network to share parameters across time, but is shallow. The output of convolution is a sequence where each member of the output is a function of a small number of neighboring members of the input. The idea of parameter sharing manifests in the application of the same convolution kernel at each time step. Recurrent networks share parameters in a diﬀerent way. Each member of the output is a function of the previous members of the output. Each member of the output is produced using the same update rule applied to the previous outputs. This recurrent formulation results in the sharing of parameters through a very deep computational graph. For the simplicity of exposition, we refer to RNNs as operating on a sequence that contains vectors x(t) with the time step index t ranging from 1 to τ . In practice, recurrent networks usually operate on minibatches of such sequences, with a diﬀerent sequence length τ for each member of the minibatch. We have omitted the minibatch indices to simplify notation. Moreover, the time step index need not literally refer to the passage of time in the real world, but only to the position in the sequence. RNNs may also be applied in two dimensions across spatial data such as images, and even when applied to data involving time, the network may have connections that go backwards in time, provided that the entire sequence is observed before it is provided to the network. This chapter extends the idea of a computational graph to include cycles. These cycles represent the inﬂuence of the present value of a variable on its own value at a future time step. Such computational graphs allow us to deﬁne recurrent neural networks. We then describe many diﬀerent ways to construct, train, and use recurrent neural networks. For more information on recurrent neural networks than is available in this chapter, we refer the reader to the textbook of Graves (2012).

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10.1

Unfolding Computational Graphs

A computational graph is a way to formalize the structure of a set of computations, such as those involved in mapping inputs and parameters to outputs and loss. Please refer to Sec. 6.5.1 for a general introduction. In this section we explain the idea of unfolding a recursive or recurrent computation into a computational graph that has a repetitive structure, typically corresponding to a chain of events. Unfolding this graph results in the sharing of parameters across a deep network structure. For example, consider the classical form of a dynamical system: s(t) = f (s(t−1) ; θ),

(10.1)

where s(t) is called the state of the system. Eq. 10.1 is recurrent because the deﬁnition of s at time t refers back to the same deﬁnition at time t − 1. For a ﬁnite number of time steps τ , the graph can be unfolded by applying the deﬁnition τ − 1 times. For example, if we unfold Eq. 10.1 for τ = 3 time steps, we obtain s(3) =f (s (2) ; θ)

(10.2)

=f (f (s (1); θ); θ)

(10.3)

Unfolding the equation by repeatedly applying the deﬁnition in this way has yielded an expression that does not involve recurrence. Such an expression can now be represented by a traditional directed acyclic computational graph. The unfolded computational graph of Eq. 10.1 and Eq. 10.3 is illustrated in Fig. 10.1. s(... )

f

s (t−1)

f

s(t)

f

s (t+1)

f

s(... )

Figure 10.1: The classical dynamical system described by Eq. 10.1, illustrated as an unfolded computational graph. Each node represents the state at some time t and the function f maps the state at t to the state at t + 1. The same parameters (the same value of θ used to parametrize f ) are used for all time steps.

As another example, let us consider a dynamical system driven by an external signal x (t), s(t) = f (s(t−1) , x(t) ; θ), (10.4) 375

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where we see that the state now contains information about the whole past sequence. Recurrent neural networks can be built in many diﬀerent ways. Much as almost any function can be considered a feedforward neural network, essentially any function involving recurrence can be considered a recurrent neural network. Many recurrent neural networks use Eq. 10.5 or a similar equation to deﬁne the values of their hidden units. To indicate that the state is the hidden units of the network, we now rewrite Eq. 10.4 using the variable h to represent the state: h(t) = f (h(t−1) , x(t) ; θ),

(10.5)

illustrated in Fig. 10.2, typical RNNs will add extra architectural features such as output layers that read information out of the state h to make predictions. When the recurrent network is trained to perform a task that requires predicting the future from the past, the network typically learns to use h(t) as a kind of lossy summary of the task-relevant aspects of the past sequence of inputs up to t. This summary is in general necessarily lossy, since it maps an arbitrary length sequence (x(t) , x(t−1) , x(t−2) , . . . , x(2) , x(1) ) to a ﬁxed length vector h(t). Depending on the training criterion, this summary might selectively keep some aspects of the past sequence with more precision than other aspects. For example, if the RNN is used in statistical language modeling, typically to predict the next word given previous words, it may not be necessary to store all of the information in the input sequence up to time t, but rather only enough information to predict the rest of the sentence. The most demanding situation is when we ask h(t) to be rich enough to allow one to approximately recover the input sequence, as in autoencoder frameworks (Chapter 14). h(t−1)

h(... )

h

f f x

h(t) f

h (t+1) f

h(... )

f

Unfold x(t−1)

x(t)

x(t+1)

Figure 10.2: A recurrent network with no outputs. This recurrent network just processes information from the input x by incorporating it into the state h that is passed forward through time. (Left) Circuit diagram. The black square indicates a delay of 1 time step. (Right) The same network seen as an unfolded computational graph, where each node is now associated with one particular time instance.

Eq. 10.5 can be drawn in two diﬀerent ways. One way to draw the RNN is with a diagram containing one node for every component that might exist in a 376

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physical implementation of the model, such as a biological neural network. In this view, the network deﬁnes a circuit that operates in real time, with physical parts whose current state can inﬂuence their future state, as in the left of Fig. 10.2. Throughout this chapter, we use a black square in a circuit diagram to indicate that an interaction takes place with a delay of 1 time step, from the state at time t to the state at time t + 1. The other way to draw the RNN is as an unfolded computational graph, in which each component is represented by many diﬀerent variables, with one variable per time step, representing the state of the component at that point in time. Each variable for each time step is drawn as a separate node of the computational graph, as in the right of Fig. 10.2. What we call unfolding is the operation that maps a circuit as in the left side of the ﬁgure to a computational graph with repeated pieces as in the right side. The unfolded graph now has a size that depends on the sequence length. We can represent the unfolded recurrence after t steps with a function g (t): h(t) =g(t) (x(t), x(t−1) , x(t−2) , . . . , x(2), x (1)) =f (h(t−1) , x(t); θ)

(10.6) (10.7)

The function g(t) takes the whole past sequence (x(t), x(t−1) , x(t−2) , . . . , x(2), x (1) ) as input and produces the current state, but the unfolded recurrent structure allows us to factorize g (t) into repeated application of a function f . The unfolding process thus introduces two major advantages: 1. Regardless of the sequence length, the learned model always has the same input size, because it is speciﬁed in terms of transition from one state to another state, rather than speciﬁed in terms of a variable-length history of states. 2. It is possible to use the same transition function f with the same parameters at every time step. These two factors make it possible to learn a single model f that operates on all time steps and all sequence lengths, rather than needing to learn a separate model g(t) for all possible time steps. Learning a single, shared model allows generalization to sequence lengths that did not appear in the training set, and allows the model to be estimated with far fewer training examples than would be required without parameter sharing. Both the recurrent graph and the unrolled graph have their uses. The recurrent graph is succinct. The unfolded graph provides an explicit description of which computations to perform. The unfolded graph also helps to illustrate the idea of 377

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information ﬂow forward in time (computing outputs and losses) and backward in time (computing gradients) by explicitly showing the path along which this information ﬂows.

10.2

Recurrent Neural Networks

Armed with the graph unrolling and parameter sharing ideas of Sec. 10.1, we can design a wide variety of recurrent neural networks. y

y (t−1)

y (t)

y (t+1)

L

L(t−1)

L(t)

L(t+1)

o

o(t−1)

o(t)

o(t+1)

Unfold

V

V

W

h(... )

h U x

W

V W

h(t−1)

h(t)

U

x(t−1)

V W

U

x(t)

W

h(t+1)

h(... )

U

x(t+1)

Figure 10.3: The computational graph to compute the training loss of a recurrent network that maps an input sequence of x values to a corresponding sequence of output o values. A loss L measures how far each o is from the corresponding training target y. When using softmax outputs, we assume o is the unnormalized log probabilities. The loss L internally ˆ = softmax(o) and compares this to the target y. The RNN has input to hidden computes y connections parametrized by a weight matrix U , hidden-to-hidden recurrent connections parametrized by a weight matrix W , and hidden-to-output connections parametrized by a weight matrix V . Eq. 10.8 deﬁnes forward propagation in this model. (Left) The RNN and its loss drawn with recurrent connections. (Right) The same seen as an time-unfolded computational graph, where each node is now associated with one particular time instance.

Some examples of important design patterns for recurrent neural networks include the following: • Recurrent networks that produce an output at each time step and have 378

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recurrent connections between hidden units, illustrated in Fig. 10.3. • Recurrent networks that produce an output at each time step and have recurrent connections only from the output at one time step to the hidden units at the next time step, illustrated in Fig. 10.4 • Recurrent networks with recurrent connections between hidden units, that read an entire sequence and then produce a single output, illustrated in Fig. 10.5. Fig. 10.3 is a reasonably representative example that we return to throughout most of the chapter. The recurrent neural network of Fig. 10.3 and Eq. 10.8 is universal in the sense that any function computable by a Turing machine can be computed by such a recurrent network of a ﬁnite size. The output can be read from the RNN after a number of time steps that is asymptotically linear in the number of time steps used by the Turing machine and asymptotically linear in the length of the input (Siegelmann and Sontag, 1991; Siegelmann, 1995; Siegelmann and Sontag, 1995; Hyotyniemi, 1996). The functions computable by a Turing machine are discrete, so these results regard exact implementation of the function, not approximations. The RNN, when used as a Turing machine, takes a binary sequence as input and its outputs must be discretized to provide a binary output. It is possible to compute all functions in this setting using a single speciﬁc RNN of ﬁnite size (Siegelmann and Sontag (1995) use 886 units). The “input” of the Turing machine is a speciﬁcation of the function to be computed, so the same network that simulates this Turing machine is suﬃcient for all problems. The theoretical RNN used for the proof can simulate an unbounded stack by representing its activations and weights with rational numbers of unbounded precision. We now develop the forward propagation equations for the RNN depicted in Fig. 10.3. The ﬁgure does not specify the choice of activation function for the hidden units. Here we assume the hyperbolic tangent activation function. Also, the ﬁgure does not specify exactly what form the output and loss function take. Here we assume that the output is discrete, as if the RNN is used to predict words or characters. A natural way to represent discrete variables is to regard the output o as giving the unnormalized log probabilities of each possible value of the discrete variable. We can then apply the softmax operation as a post-processing step to obtain a vector yˆ of normalized probabilities over the output. Forward propagation begins with a speciﬁcation of the initial state h(0) . Then, for each time step from t = 1 to t = τ , we apply the following update equations: a(t)

= b + W h(t−1) + U x(t) 379

(10.8)

CHAPTER 10. SEQUENCE MODELING: RECURRENT AND RECURSIVE NETS

y

y (t−1)

y (t)

y (t+1)

L

L(t−1)

L(t)

L(t+1)

o(t−1)

o(t)

o(t+1)

o

o(... )

W

V

W V

W V

W V

W Unfold

h

U x

h(t−1)

U x(t−1)

h(t)

U x(t)

h(t+1)

h(... )

U x(t+1)

Figure 10.4: An RNN whose only recurrence is the feedback connection from the output to the hidden layer. At each time step t, the input is x t, the hidden layer activations are h(t) , the outputs are o(t), the targets are y (t) and the loss is L(t) . (Left) Circuit diagram. (Right) Unfolded computational graph. Such an RNN is less powerful (can express a smaller set of functions) than those in the family represented by Fig. 10.3. The RNN in Fig. 10.3 can choose to put any information it wants about the past into its hidden representation h and transmit h to the future. The RNN in this ﬁgure is trained to put a speciﬁc output value into o, and o is the only information it is allowed to send to the future. There are no direct connections from h going forward. The previous h is connected to the present only indirectly, via the predictions it was used to produce. Unless o is very high-dimensional and rich, it will usually lack important information from the past. This makes the RNN in this ﬁgure less powerful, but it may be easier to train because each time step can be trained in isolation from the others, allowing greater parallelization during training, as described in Sec. 10.2.1.

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h(t)

= tanh(a (t))

(t)

(t)

o

(t)

yˆ

= c+Vh

(10.9) (10.10) (t)

= softmax(o )

(10.11)

where the parameters are the bias vectors b and c along with the weight matrices U, V and W , respectively for input-to-hidden, hidden-to-output and hidden-tohidden connections. This is an example of a recurrent network that maps an input sequence to an output sequence of the same length. The total loss for a given sequence of x values paired with a sequence of y values would then be just the sum of the losses over all the time steps. For example, if L(t) is the negative log-likelihood of y(t) given x (1) , . . . , x(t) , then L {x(1), . . . , x (τ ) }, {y(1) , . . . , y (τ ) } (10.12) = L(t) (10.13) t

=−

t

log p model y (t) | {x(1), . . . , x (t)} ,

(10.14)

where pmodel y (t) | {x(1) , . . . , x(t) } is given by reading the entry for y (t) from the model’s output vector yˆ(t) . Computing the gradient of this loss function with respect to the parameters is an expensive operation. The gradient computation involves performing a forward propagation pass moving left to right through our illustration of the unrolled graph in Fig. 10.3, followed by a backward propagation pass moving right to left through the graph. The runtime is O(τ) and cannot be reduced by parallelization because the forward propagation graph is inherently sequential; each time step may only be computed after the previous one. States computed in the forward pass must be stored until they are reused during the backward pass, so the memory cost is also O(τ ). The back-propagation algorithm applied to the unrolled graph with O(τ ) cost is called back-propagation through time or BPTT and is discussed further Sec. 10.2.2. The network with recurrence between hidden units is thus very powerful but also expensive to train. Is there an alternative?

10.2.1

Teacher Forcing and Networks with Output Recurrence

The network with recurrent connections only from the output at one time step to the hidden units at the next time step (shown in Fig. 10.4) is strictly less powerful because it lacks hidden-to-hidden recurrent connections. For example, it cannot simulate a universal Turing machine. Because this network lacks hidden-to-hidden 381

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recurrence, it requires that the output units capture all of the information about the past that the network will use to predict the future. Because the output units are explicitly trained to match the training set targets, they are unlikely to capture the necessary information about the past history of the input, unless the user knows how to describe the full state of the system and provides it as part of the training set targets. The advantage of eliminating hidden-to-hidden recurrence is that, for any loss function based on comparing the prediction at time t to the training target at time t, all the time steps are decoupled. Training can thus be parallelized, with the gradient for each step t computed in isolation. There is no need to compute the output for the previous time step ﬁrst, because the training set provides the ideal value of that output. L(τ )

y (τ )

o(τ )

V ...

W

h(t−1) W

U x(t−1)

h(t)

U

...

W

W U

x(t)

x(...)

h(τ )

U x (τ )

Figure 10.5: Time-unfolded recurrent neural network with a single output at the end of the sequence. Such a network can be used to summarize a sequence and produce a ﬁxed-size representation used as input for further processing. There might be a target right at the end (as depicted here) or the gradient on the output o(t) can be obtained by back-propagating from further downstream modules.

Models that have recurrent connections from their outputs leading back into the model may be trained with teacher forcing. Teacher forcing is a procedure that emerges from the maximum likelihood criterion, in which during training the model receives the ground truth output y(t) as input at time t + 1. We can see this by examining a sequence with two time steps. The conditional maximum likelihood criterion is (1) (2) (1) (2) log p y , y | x , x (10.15) 382

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y (t−1)

y (t)

L (t−1)

L (t)

W o(t−1)

o(t−1)

o(t)

o(t)

W V

V

V

h (t−1)

h (t)

h(t−1)

U

U

U

x(t−1)

x(t−1)

x(t)

Train time

V h(t)

U x(t)

Test time

Figure 10.6: Illustration of teacher forcing. Teacher forcing is a training technique that is applicable to RNNs that have connections from their output to their hidden states at the next time step. (Left) At train time, we feed the correct output y(t) drawn from the train set as input to h(t+1) . (Right) When the model is deployed, the true output is generally not known. In this case, we approximate the correct output y(t) with the model’s output o (t), and feed the output back into the model.

383

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= log p y

(2)

|y

(1)

,x

(1)

,x

(2)

(1) (1) (2) + log p y | x , x

(10.16)

In this example, we see that at time t = 2, the model is trained to maximize the conditional probability of y (2) given both the x sequence so far and the previous y value from the training set. Maximum likelihood thus speciﬁes that during training, rather than feeding the model’s own output back into itself, these connections should be fed with the target values specifying what the correct output should be. This is illustrated in Fig. 10.6. We originally motivated teacher forcing as allowing us to avoid back-propagation through time in models that lack hidden-to-hidden connections. Teacher forcing may still be applied to models that have hidden-to-hidden connections so long as they have connections from the output at one time step to values computed in the next time step. However, as soon as the hidden units become a function of earlier time steps, the BPTT algorithm is necessary. Some models may thus be trained with both teacher forcing and BPTT. The disadvantage of strict teacher forcing arises if the network is going to be later used in an open-loop mode, with the network outputs (or samples from the output distribution) fed back as input. In this case, the kind of inputs that the network sees during training could be quite diﬀerent from the kind of inputs that it will see at test time. One way to mitigate this problem is to train with both teacher-forced inputs and with free-running inputs, for example by predicting the correct target a number of steps in the future through the unfolded recurrent output-to-input paths. In this way, the network can learn to take into account input conditions (such as those it generates itself in the free-running mode) not seen during training and how to map the state back towards one that will make the network generate proper outputs after a few steps. Another approach (Bengio et al., 2015b) to mitigate the gap between the inputs seen at train time and the inputs seen at test time randomly chooses to use generated values or actual data values as input. This approach exploits a curriculum learning strategy to gradually use more of the generated values as input.

10.2.2

Computing the Gradient in a Recurrent Neural Network

Computing the gradient through a recurrent neural network is straightforward. One simply applies the generalized back-propagation algorithm of Sec. 6.5.6 to the unrolled computational graph. No specialized algorithms are necessary. The use of back-propagation on the unrolled graph is called the back-propagation through time (BPTT) algorithm. Gradients obtained by back-propagation may then be used with any general-purpose gradient-based techniques to train an RNN. 384

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To gain some intuition for how the BPTT algorithm behaves, we provide an example of how to compute gradients by BPTT for the RNN equations above (Eq. 10.8 and Eq. 10.12). The nodes of our computational graph include the parameters U, V , W , b and c as well as the sequence of nodes indexed by t for x(t), h(t) , o(t) and L(t) . For each node N we need to compute the gradient ∇N L recursively, based on the gradient computed at nodes that follow it in the graph. We start the recursion with the nodes immediately preceding the ﬁnal loss ∂L = 1. ∂L(t)

(10.17)

In this derivation we assume that the outputs o(t) are used as the argument to the softmax function to obtain the vector yˆ of probabilities over the output. We also assume that the loss is the negative log-likelihood of the true target y(t) given the input so far. The gradient ∇ o(t) L on the outputs at time step t, for all i, t, is as follows: ∂L ∂L ∂L(t) (t) = y ˆ (10.18) (∇o(t) L)i = (t) = i − 1 i,y(t) . ∂L(t) ∂o(i t) ∂oi We work our way backwards, starting from the end of the sequence. At the ﬁnal time step τ , h (τ ) only has o(τ ) as a descendent, so its gradient is simple: ∇h(τ ) L = V ∇o(τ ) L.

(10.19)

We can then iterate backwards in time to back-propagate gradients through time, from t = τ − 1 down to t = 1, noting that h(t) (for t < τ ) has as descendents both o (t) and h(t+1) . Its gradient is thus given by

∂o(t) ∇h (t) L = (∇h(t+1) L) + (∇o(t) L) ∂h (t) 2 (t+1) = W (∇ h(t+1)L) diag 1 − h + V (∇ o(t) L) ∂h(t+1) ∂h(t)

(10.20) (10.21)

(t+1) 2 where diag 1 − h indicates the diagonal matrix containing the elements (t+1)

1 − (hi )2 . This is the Jacobian of the hyperbolic tangent associated with the hidden unit i at time t + 1. Once the gradients on the internal nodes of the computational graph are obtained, we can obtain the gradients on the parameter nodes. Because the parameters are shared across many time steps, we must take some care when denoting calculus operations involving these variables. The equations we wish to 385

CHAPTER 10. SEQUENCE MODELING: RECURRENT AND RECURSIVE NETS

implement use the bprop method of Sec. 6.5.6, that computes the contribution of a single edge in the computational graph to the gradient. However, the ∇ W f operator used in calculus takes into account the contribution of W to the value of f due to all edges in the computational graph. To resolve this ambiguity, we introduce dummy variables W (t) that are deﬁned to be copies of W but with each W (t) used only at time step t. We may then use ∇ W (t) to denote the contribution of the weights at time step t to the gradient. Using this notation, the gradient on the remaining parameters is given by: ∂o(t) ∇cL = ∇o(t) L = ∇ o(t) L (10.22) ∂c t t 2 ∂h(t) (t) diag 1 − h ∇bL = ∇h (t)L = ∇h(t) L (10.23) (t) ∂b t t ∂L (t) (t) o = (10.24) ∇V L = ∇ (∇ (t) L) h V i o (t) ∂o t t i i ∂L (t) (10.25) ∇W L = ∇ W (t)h i (t) ∂h t i i 2 (t) = diag 1 − h (10.26) (∇h (t) L) h(t−1) t

∇U L = =

t

t

i

∂L

∂h(i t)

∇ U(t) h (i t)

2 (t) diag 1 − h (∇h (t) L) x(t)

(10.27) (10.28)

We do not need to compute the gradient with respect to x(t) for training because it does not have any parameters as ancestors in the computational graph deﬁning the loss.

10.2.3

Recurrent Networks as Directed Graphical Models

In the example recurrent network we have developed so far, the losses L(t) were cross-entropies between training targets y(t) and outputs o(t) . As with a feedforward network, it is in principle possible to use almost any loss with a recurrent network. The loss should be chosen based on the task. As with a feedforward network, we usually wish to interpret the output of the RNN as a probability distribution, and 386

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we usually use the cross-entropy associated with that distribution to deﬁne the loss. Mean squared error is the cross-entropy loss associated with an output distribution that is a unit Gaussian, for example, just as with a feedforward network. When we use a predictive log-likelihood training objective, such as Eq. 10.12, we train the RNN to estimate the conditional distribution of the next sequence element y(t) given the past inputs. This may mean that we maximize the log-likelihood log p(y(t) | x(1) , . . . , x (t)),

(10.29)

or, if the model includes connections from the output at one time step to the next time step, log p(y(t) | x (1) , . . . , x(t), y (1) , . . . , y(t−1) ). (10.30) Decomposing the joint probability over the sequence of y values as a series of one-step probabilistic predictions is one way to capture the full joint distribution across the whole sequence. When we do not feed past y values as inputs that condition the next step prediction, the directed graphical model contains no edges from any y(i) in the past to the current y(t) . In this case, the outputs y are conditionally independent given the sequence of x values. When we do feed the actual y values (not their prediction, but the actual observed or generated values) back into the network, the directed graphical model contains edges from all y(i) values in the past to the current y(t) value.

y (1)

y (2)

y (3)

y (4)

y (5)

y (...)

Figure 10.7: Fully connected graphical model for a sequence y (1) , y (2), . . . , y (t) , . . . : every past observation y(i) may inﬂuence the conditional distribution of some y (t) (for t > i), given the previous values. Parametrizing the graphical model directly according to this graph (as in Eq. 10.6) might be very ineﬃcient, with an ever growing number of inputs and parameters for each element of the sequence. RNNs obtain the same full connectivity but eﬃcient parametrization, as illustrated in Fig. 10.8. 387

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As a simple example, let us consider the case where the RNN models only a sequence of scalar random variables Y = {y(1) , . . . , y (τ )}, with no additional inputs x. The input at time step t is simply the output at time step t − 1. The RNN then deﬁnes a directed graphical model over the y variables. We parametrize the joint distribution of these observations using the chain rule (Eq. 3.6) for conditional probabilities: P (Y) = P (y

(1)

(τ )

,...,y

)=

τ t=1

P (y (t) | y(t−1) , y(t−2) , . . . , y(1) )

(10.31)

where the right-hand side of the bar is empty for t = 1, of course. Hence the negative log-likelihood of a set of values {y (1) , . . . , y (τ )} according to such a model is L= L(t) (10.32) t

where

L (t) = − log P (y(t) = y (t) | y(t−1) , y (t−2) , . . . , y (1)). h(1)

h(2)

h(3)

h(4)

h(5)

h(... )

y (1)

y (2)

y (3)

y (4)

y (5)

y (...)

(10.33)

Figure 10.8: Introducing the state variable in the graphical model of the RNN, even though it is a deterministic function of its inputs, helps to see how we can obtain a very eﬃcient parametrization, based on Eq. 10.5. Every stage in the sequence (for h(t) and y(t) ) involves the same structure (the same number of inputs for each node) and can share the same parameters with the other stages.

The edges in a graphical model indicate which variables depend directly on other variables. Many graphical models aim to achieve statistical and computational eﬃciency by omitting edges that do not correspond to strong interactions. For example, it is common to make the Markov assumption that the graphical model should only contain edges from {y(t−k) , . . . , y(t−1)} to y(t) , rather than containing edges from the entire past history. However, in some cases, we believe that all past inputs should have an inﬂuence on the next element of the sequence. RNNs are useful when we believe that the distribution over y(t) may depend on a value of y(i) from the distant past in a way that is not captured by the eﬀect of y(i) on y (t−1). 388

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One way to interpret an RNN as a graphical model is to view the RNN as deﬁning a graphical model whose structure is the complete graph, able to represent direct dependencies between any pair of y values. The graphical model over the y values with the complete graph structure is shown in Fig. 10.7. The complete graph interpretation of the RNN is based on ignoring the hidden units h(t) by marginalizing them out of the model. It is more interesting to consider the graphical model structure of RNNs that results from regarding the hidden units h(t) as random variables. 1 Including the hidden units in the graphical model reveals that the RNN provides a very eﬃcient parametrization of the joint distribution over the observations. Suppose that we represented an arbitrary joint distribution over discrete values with a tabular representation—an array containing a separate entry for each possible assignment of values, with the value of that entry giving the probability of that assignment occurring. If y can take on k diﬀerent values, the tabular representation would have O(kτ ) parameters. By comparison, due to parameter sharing, the number of parameters in the RNN is O(1) as a function of sequence length. The number of parameters in the RNN may be adjusted to control model capacity but is not forced to scale with sequence length. Eq. 10.5 shows that the RNN parametrizes long-term relationships between variables eﬃciently, using recurrent applications of the same function f and same parameters θ at each time step. Fig. 10.8 illustrates the graphical model interpretation. Incorporating the h(t) nodes in the graphical model decouples the past and the future, acting as an intermediate quantity between them. A variable y (i) in the distant past may inﬂuence a variable y(t) via its eﬀect on h. The structure of this graph shows that the model can be eﬃciently parametrized by using the same conditional probability distributions at each time step, and that when the variables are all observed, the probability of the joint assignment of all variables can be evaluated eﬃciently. Even with the eﬃcient parametrization of the graphical model, some operations remain computationally challenging. For example, it is diﬃcult to predict missing values in the middle of the sequence. The price recurrent networks pay for their reduced number of parameters is that optimizing the parameters may be diﬃcult. The parameter sharing used in recurrent networks relies on the assumption that the same parameters can be used for diﬀerent time steps. Equivalently, the assumption is that the conditional probability distribution over the variables at 1

The conditional distribution over these variables given their parents is deterministic. This is perfectly legitimate, though it is somewhat rare to design a graphical model with such deterministic hidden units. 389

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time t + 1 given the variables at time t is stationary, meaning that the relationship between the previous time step and the next time step does not depend on t. In principle, it would be possible to use t as an extra input at each time step and let the learner discover any time-dependence while sharing as much as it can between diﬀerent time steps. This would already be much better than using a diﬀerent conditional probability distribution for each t, but the network would then have to extrapolate when faced with new values of t. To complete our view of an RNN as a graphical model, we must describe how to draw samples from the model. The main operation that we need to perform is simply to sample from the conditional distribution at each time step. However, there is one additional complication. The RNN must have some mechanism for determining the length of the sequence. This can be achieved in various ways. In the case when the output is a symbol taken from a vocabulary, one can add a special symbol corresponding to the end of a sequence (Schmidhuber, 2012). When that symbol is generated, the sampling process stops. In the training set, we insert this symbol as an extra member of the sequence, immediately after x(τ ) in each training example. Another option is to introduce an extra Bernoulli output to the model that represents the decision to either continue generation or halt generation at each time step. This approach is more general than the approach of adding an extra symbol to the vocabulary, because it may be applied to any RNN, rather than only RNNs that output a sequence of symbols. For example, it may be applied to an RNN that emits a sequence of real numbers. The new output unit is usually a sigmoid unit trained with the cross-entropy loss. In this approach the sigmoid is trained to maximize the log-probability of the correct prediction as to whether the sequence ends or continues at each time step. Another way to determine the sequence length τ is to add an extra output to the model that predicts the integer τ itself. The model can sample a value of τ and then sample τ steps worth of data. This approach requires adding an extra input to the recurrent update at each time step so that the recurrent update is aware of whether it is near the end of the generated sequence. This extra input can either consist of the value of τ or can consist of τ − t , the number of remaining time steps. Without this extra input, the RNN might generate sequences that end abruptly, such as a sentence that ends before it is complete. This approach is based on the decomposition P (x (1), . . . , x(τ )) = P (τ )P (x (1) , . . . , x(τ ) | τ ).

(10.34)

The strategy of predicting τ directly is used for example by Goodfellow et al. 390

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(2014d).

10.2.4

Modeling Sequences Conditioned on Context with RNNs

In the previous section we described how an RNN could correspond to a directed graphical model over a sequence of random variables y (t) with no inputs x. Of course, our development of RNNs as in Eq. 10.8 included a sequence of inputs x(1) , x(2) , . . . , x(τ ) . In general, RNNs allow the extension of the graphical model view to represent not only a joint distribution over the y variables but also a conditional distribution over y given x. As discussed in the context of feedforward networks in Sec. 6.2.1.1, any model representing a variable P (y; θ) can be reinterpreted as a model representing a conditional distribution P (y|ω ) with ω = θ . We can extend such a model to represent a distribution P ( y | x) by using the same P(y | ω ) as before, but making ω a function of x. In the case of an RNN, this can be achieved in diﬀerent ways. We review here the most common and obvious choices. Previously, we have discussed RNNs that take a sequence of vectors x(t) for t = 1, . . . , τ as input. Another option is to take only a single vector x as input. When x is a ﬁxed-size vector, we can simply make it an extra input of the RNN that generates the y sequence. Some common ways of providing an extra input to an RNN are: 1. as an extra input at each time step, or 2. as the initial state h(0), or 3. both. The ﬁrst and most common approach is illustrated in Fig. 10.9. The interaction between the input x and each hidden unit vector h(t) is parametrized by a newly introduced weight matrix R that was absent from the model of only the sequence of y values. The same product xR is added as additional input to the hidden units at every time step. We can think of the choice of x as determining the value of x R that is eﬀectively a new bias parameter used for each of the hidden units. The weights remain independent of the input. We can think of this model as taking the parameters θ of the non-conditional model and turning them into ω, where the bias parameters within ω are now a function of the input. Rather than receiving only a single vector x as input, the RNN may receive a sequence of vectors x(t) as input. The RNN described in Eq. 10.8 corresponds to a 391

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y (t−1)

U

y (t)

U

L(t−1)

o(t−1)

W

W

R

L(t+1)

V W

h(t)

R

y (...)

o(t+1)

V

h(t−1)

R

U

L(t)

o(t)

V s(... )

y (t+1)

W h(t+1)

R

h(... )

R

x

Figure 10.9: An RNN that maps a ﬁxed-length vector x into a distribution over sequences Y. This RNN is appropriate for tasks such as image captioning, where a single image is used as input to a model that then produces a sequence of words describing the image. Each element y (t) of the observed output sequence serves both as input (for the current time step) and, during training, as target (for the previous time step).

392

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y (t−1)

y (t)

y (t+1)

L(t−1)

L(t)

L(t+1)

R o(t−1)

R o(t)

V W (... )

h

h

U x(t−1)

o(t+1)

V W

(t−1)

R

V W

(t)

h

U x(t)

W (t+1)

h

h(... )

U x(t+1)

Figure 10.10: A conditional recurrent neural network mapping a variable-length sequence of x values into a distribution over sequences of y values of the same length. Compared to Fig. 10.3, this RNN contains connections from the previous output to the current state. These connections allow this RNN to model an arbitrary distribution over sequences of y given sequences of x of the same length. The RNN of Fig. 10.3 is only able to represent distributions in which the y values are conditionally independent from each other given the x values.

393

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conditional distribution P (y(1) , . . . , y(τ ) | x(1) , . . . , x(τ ) ) that makes a conditional independence assumption that this distribution factorizes as P (y (t) | x(1) , . . . , x(t)). (10.35) t

To remove the conditional independence assumption, we can add connections from the output at time t to the hidden unit at time t + 1, as shown in Fig. 10.10. The model can then represent arbitrary probability distributions over the y sequence. This kind of model representing a distribution over a sequence given another sequence still has one restriction, which is that the length of both sequences must be the same. We describe how to remove this restriction in Sec. 10.4. y (t−1)

y (t)

y (t+1)

L(t−1)

L(t)

L(t+1)

o(t−1)

o(t)

o(t+1)

g (t−1)

g (t)

g (t+1)

h(t−1)

h(t)

h(t+1)

x(t−1)

x(t)

x (t+1)

Figure 10.11: Computation of a typical bidirectional recurrent neural network, meant to learn to map input sequences x to target sequences y , with loss L(t) at each step t. The h recurrence propagates information forward in time (towards the right) while the g recurrence propagates information backward in time (towards the left). Thus at each point t, the output units o(t) can beneﬁt from a relevant summary of the past in its h(t) input and from a relevant summary of the future in its g (t) input.

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10.3

Bidirectional RNNs

All of the recurrent networks we have considered up to now have a “causal” structure, meaning that the state at time t only captures information from the past, x(1) , . . . , x (t−1), and the present input x(t). Some of the models we have discussed also allow information from past y values to aﬀect the current state when the y values are available. However, in many applications we want to output a prediction of y(t) which may depend on the whole input sequence. For example, in speech recognition, the correct interpretation of the current sound as a phoneme may depend on the next few phonemes because of co-articulation and potentially may even depend on the next few words because of the linguistic dependencies between nearby words: if there are two interpretations of the current word that are both acoustically plausible, we may have to look far into the future (and the past) to disambiguate them. This is also true of handwriting recognition and many other sequence-to-sequence learning tasks, described in the next section. Bidirectional recurrent neural networks (or bidirectional RNNs) were invented to address that need (Schuster and Paliwal, 1997). They have been extremely successful (Graves, 2012) in applications where that need arises, such as handwriting recognition (Graves et al., 2008; Graves and Schmidhuber, 2009), speech recognition (Graves and Schmidhuber, 2005; Graves et al., 2013) and bioinformatics (Baldi et al., 1999). As the name suggests, bidirectional RNNs combine an RNN that moves forward through time beginning from the start of the sequence with another RNN that moves backward through time beginning from the end of the sequence. Fig. 10.11 illustrates the typical bidirectional RNN, with h(t) standing for the state of the sub-RNN that moves forward through time and g(t) standing for the state of the sub-RNN that moves backward through time. This allows the output units o(t) to compute a representation that depends on both the past and the future but is most sensitive to the input values around time t, without having to specify a ﬁxed-size window around t (as one would have to do with a feedforward network, a convolutional network, or a regular RNN with a ﬁxed-size look-ahead buﬀer). This idea can be naturally extended to 2-dimensional input, such as images, by having four RNNs, each one going in one of the four directions: up, down, left, right. At each point (i, j) of a 2-D grid, an output Oi,j could then compute a representation that would capture mostly local information but could also depend on long-range inputs, if the RNN is able to learn to carry that information. Compared to a convolutional network, RNNs applied to images are typically more 395

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expensive but allow for long-range lateral interactions between features in the same feature map (Visin et al., 2015; Kalchbrenner et al., 2015). Indeed, the forward propagation equations for such RNNs may be written in a form that shows they use a convolution that computes the bottom-up input to each layer, prior to the recurrent propagation across the feature map that incorporates the lateral interactions.

10.4

Encoder-Decoder Sequence-to-Sequence Architectures

We have seen in Fig. 10.5 how an RNN can map an input sequence to a ﬁxed-size vector. We have seen in Fig. 10.9 how an RNN can map a ﬁxed-size vector to a sequence. We have seen in Fig. 10.3, Fig. 10.4, Fig. 10.10 and Fig. 10.11 how an RNN can map an input sequence to an output sequence of the same length. Here we discuss how an RNN can be trained to map an input sequence to an output sequence which is not necessarily of the same length. This comes up in many applications, such as speech recognition, machine translation or question answering, where the input and output sequences in the training set are generally not of the same length (although their lengths might be related). We often call the input to the RNN the “context.” We want to produce a representation of this context, C . The context C might be a vector or sequence of vectors that summarize the input sequence X = (x(1) , . . . , x (nx ) ). The simplest RNN architecture for mapping a variable-length sequence to another variable-length sequence was ﬁrst proposed by Cho et al. (2014a) and shortly after by Sutskever et al. (2014), who independently developed that architecture and were the ﬁrst to obtain state-of-the-art translation using this approach. The former system is based on scoring proposals generated by another machine translation system, while the latter uses a standalone recurrent network to generate the translations. These authors respectively called this architecture, illustrated in Fig. 10.12, the encoder-decoder or sequence-to-sequence architecture. The idea is very simple: (1) an encoder or reader or input RNN processes the input sequence. The encoder emits the context C, usually as a simple function of its ﬁnal hidden state. (2) a decoder or writer or output RNN is conditioned on that ﬁxed-length vector (just like in Fig. 10.9) to generate the output sequence Y = (y(1) , . . . , y (ny ) ). The innovation of this kind of architecture over those presented in earlier sections of this chapter is that the lengths nx and ny can vary from each other, while previous architectures constrained nx = ny = τ. In a sequence-to-sequence architecture, the two RNNs 396

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Encoder …

x(1)

x(2)

x(...)

x(n x )

C

Decoder …

y (1)

y (2)

y (...)

y (n y )

Figure 10.12: Example of an encoder-decoder or sequence-to-sequence RNN architecture, for learning to generate an output sequence (y (1) , . . . , y(n y) ) given an input sequence (x(1) , x(2) , . . . , x(nx) ). It is composed of an encoder RNN that reads the input sequence and a decoder RNN that generates the output sequence (or computes the probability of a given output sequence). The ﬁnal hidden state of the encoder RNN is used to compute a generally ﬁxed-size context variable C which represents a semantic summary of the input sequence and is given as input to the decoder RNN.

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are trained jointly to maximize the average of log P (y(1) , . . . , y (ny) | x (1) , . . . , x(nx ) ) over all the pairs of x and y sequences in the training set. The last state hn x of the encoder RNN is typically used as a representation C of the input sequence that is provided as input to the decoder RNN. If the context C is a vector, then the decoder RNN is simply a vector-tosequence RNN as described in Sec. 10.2.4. As we have seen, there are at least two ways for a vector-to-sequence RNN to receive input. The input can be provided as the initial state of the RNN, or the input can be connected to the hidden units at each time step. These two ways can also be combined. There is no constraint that the encoder must have the same size of hidden layer as the decoder. One clear limitation of this architecture is when the context C output by the encoder RNN has a dimension that is too small to properly summarize a long sequence. This phenomenon was observed by Bahdanau et al. (2015) in the context of machine translation. They proposed to make C a variable-length sequence rather than a ﬁxed-size vector. Additionally, they introduced an attention mechanism that learns to associate elements of the sequence C to elements of the output sequence. See Sec. 12.4.5.1 for more details.

10.5

Deep Recurrent Networks

The computation in most RNNs can be decomposed into three blocks of parameters and associated transformations: 1. from the input to the hidden state, 2. from the previous hidden state to the next hidden state, and 3. from the hidden state to the output. With the RNN architecture of Fig. 10.3, each of these three blocks is associated with a single weight matrix. In other words, when the network is unfolded, each of these corresponds to a shallow transformation. By a shallow transformation, we mean a transformation that would be represented by a single layer within a deep MLP. Typically this is a transformation represented by a learned aﬃne transformation followed by a ﬁxed nonlinearity. Would it be advantageous to introduce depth in each of these operations? Experimental evidence (Graves et al., 2013; Pascanu et al., 2014a) strongly suggests so. The experimental evidence is in agreement with the idea that we need enough 398

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y

y

h

h

x

x

x

(a)

(b)

(c)

y

z

h

Figure 10.13: A recurrent neural network can be made deep in many ways (Pascanu et al., 2014a). (a) The hidden recurrent state can be broken down into groups organized hierarchically. (b) Deeper computation (e.g., an MLP) can be introduced in the input-tohidden, hidden-to-hidden and hidden-to-output parts. This may lengthen the shortest path linking diﬀerent time steps. (c) The path-lengthening eﬀect can be mitigated by introducing skip connections.

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depth in order to perform the required mappings. See also Schmidhuber (1992), El Hihi and Bengio (1996), or Jaeger (2007a) for earlier work on deep RNNs. Graves et al. (2013) were the ﬁrst to show a signiﬁcant beneﬁt of decomposing the state of an RNN into multiple layers as in Fig. 10.13 (left). We can think of the lower layers in the hierarchy depicted in Fig. 10.13a as playing a role in transforming the raw input into a representation that is more appropriate, at the higher levels of the hidden state. Pascanu et al. (2014a) go a step further and propose to have a separate MLP (possibly deep) for each of the three blocks enumerated above, as illustrated in Fig. 10.13b. Considerations of representational capacity suggest to allocate enough capacity in each of these three steps, but doing so by adding depth may hurt learning by making optimization diﬃcult. In general, it is easier to optimize shallower architectures, and adding the extra depth of Fig. 10.13b makes the shortest path from a variable in time step t to a variable in time step t + 1 become longer. For example, if an MLP with a single hidden layer is used for the state-to-state transition, we have doubled the length of the shortest path between variables in any two diﬀerent time steps, compared with the ordinary RNN of Fig. 10.3. However, as argued by Pascanu et al. (2014a), this can be mitigated by introducing skip connections in the hidden-to-hidden path, as illustrated in Fig. 10.13c.

10.6

Recursive Neural Networks

Recursive neural networks2 represent yet another generalization of recurrent networks, with a diﬀerent kind of computational graph, which is structured as a deep tree, rather than the chain-like structure of RNNs. The typical computational graph for a recursive network is illustrated in Fig. 10.14. Recursive neural networks were introduced by Pollack (1990) and their potential use for learning to reason was described by by Bottou (2011). Recursive networks have been successfully applied to processing data structures as input to neural nets (Frasconi et al., 1997, 1998), in natural language processing (Socher et al., 2011a,c, 2013a) as well as in computer vision (Socher et al., 2011b). One clear advantage of recursive nets over recurrent nets is that for a sequence of the same length τ, the depth (measured as the number of compositions of nonlinear operations) can be drastically reduced from τ to O(log τ ), which might help deal with long-term dependencies. An open question is how to best structure the tree. One option is to have a tree structure which does not depend on the data, 2

We suggest to not abbreviate “recursive neural network” as “RNN” to avoid confusion with “recurrent neural network.” 400

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L

y

o

U

U

V x (1)

W

W

U

V

V

x(2)

x(3)

W

V x(4)

Figure 10.14: A recursive network has a computational graph that generalizes that of the recurrent network from a chain to a tree. A variable-size sequence x(1), x (2) , . . . , x(t) can be mapped to a ﬁxed-size representation (the output o), with a ﬁxed set of parameters (the weight matrices U , V , W ). The ﬁgure illustrates a supervised learning case in which some target y is provided which is associated with the whole sequence.

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such as a balanced binary tree. In some application domains, external methods can suggest the appropriate tree structure. For example, when processing natural language sentences, the tree structure for the recursive network can be ﬁxed to the structure of the parse tree of the sentence provided by a natural language parser (Socher et al., 2011a, 2013a). Ideally, one would like the learner itself to discover and infer the tree structure that is appropriate for any given input, as suggested by Bottou (2011). Many variants of the recursive net idea are possible. For example, Frasconi et al. (1997) and Frasconi et al. (1998) associate the data with a tree structure, and associate the inputs and targets with individual nodes of the tree. The computation performed by each node does not have to be the traditional artiﬁcial neuron computation (aﬃne transformation of all inputs followed by a monotone nonlinearity). For example, Socher et al. (2013a) propose using tensor operations and bilinear forms, which have previously been found useful to model relationships between concepts (Weston et al., 2010; Bordes et al., 2012) when the concepts are represented by continuous vectors (embeddings).

10.7

The Challenge of Long-Term Dependencies

The mathematical challenge of learning long-term dependencies in recurrent networks was introduced in Sec. 8.2.5. The basic problem is that gradients propagated over many stages tend to either vanish (most of the time) or explode (rarely, but with much damage to the optimization). Even if we assume that the parameters are such that the recurrent network is stable (can store memories, with gradients not exploding), the diﬃculty with long-term dependencies arises from the exponentially smaller weights given to long-term interactions (involving the multiplication of many Jacobians) compared to short-term ones. Many other sources provide a deeper treatment (Hochreiter, 1991; Doya, 1993; Bengio et al., 1994; Pascanu et al., 2013a) . In this section, we describe the problem in more detail. The remaining sections describe approaches to overcoming the problem. Recurrent networks involve the composition of the same function multiple times, once per time step. These compositions can result in extremely nonlinear behavior, as illustrated in Fig. 10.15. In particular, the function composition employed by recurrent neural networks somewhat resembles matrix multiplication. We can think of the recurrence relation h(t) = W h(t−1)

(10.36)

as a very simple recurrent neural network lacking a nonlinear activation function, 402

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Figure 10.15: When composing many nonlinear functions (like the linear-tanh layer shown here), the result is highly nonlinear, typically with most of the values associated with a tiny derivative, some values with a large derivative, and many alternations between increasing and decreasing. In this plot, we plot a linear projection of a 100-dimensional hidden state down to a single dimension, plotted on the y-axis. The x-axis is the coordinate of the initial state along a random direction in the 100-dimensional space. We can thus view this plot as a linear cross-section of a high-dimensional function. The plots show the function after each time step, or equivalently, after each number of times the transition function has been composed.

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and lacking inputs x. As described in Sec. 8.2.5, this recurrence relation essentially describes the power method. It may be simpliﬁed to h(t) = W t h(0) ,

(10.37)

and if W admits an eigendecomposition of the form W = QΛQ ,

(10.38)

with orthogonal Q, the recurrence may be simpliﬁed further to h(t) = Q Λt Qh(0) .

(10.39)

The eigenvalues are raised to the power of t causing eigenvalues with magnitude less than one to decay to zero and eigenvalues with magnitude greater than one to explode. Any component of h(0) that is not aligned with the largest eigenvector will eventually be discarded. This problem is particular to recurrent networks. In the scalar case, imagine multiplying a weight w by itself many times. The product wt will either vanish or explode depending on the magnitude of w. However, if we make a non-recurrent network that has a diﬀerent weight w (t) at each time step, the situation diﬀerent. (tis ) If the initial state is given by 1, then the state at time t is given by t w . Suppose that the w(t) values are generated randomly, independently from one another, with zero mean and variance v. The variance of the product is O(v n). To obtain some √ desired variance v ∗ we may choose the individual weights with variance v = n v ∗. Very deep feedforward networks with carefully chosen scaling can thus avoid the vanishing and exploding gradient problem, as argued by Sussillo (2014). The vanishing and exploding gradient problem for RNNs was independently discovered by separate researchers (Hochreiter, 1991; Bengio et al., 1993, 1994). One may hope that the problem can be avoided simply by staying in a region of parameter space where the gradients do not vanish or explode. Unfortunately, in order to store memories in a way that is robust to small perturbations, the RNN must enter a region of parameter space where gradients vanish (Bengio et al., 1993, 1994). Speciﬁcally, whenever the model is able to represent long term dependencies, the gradient of a long term interaction has exponentially smaller magnitude than the gradient of a short term interaction. It does not mean that it is impossible to learn, but that it might take a very long time to learn long-term dependencies, because the signal about these dependencies will tend to be hidden by the smallest ﬂuctuations arising from short-term dependencies. In practice, the experiments in Bengio et al. (1994) show that as we increase the span of the dependencies that 404

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need to be captured, gradient-based optimization becomes increasingly diﬃcult, with the probability of successful training of a traditional RNN via SGD rapidly reaching 0 for sequences of only length 10 or 20. For a deeper treatment of recurrent networks as dynamical systems, see Doya (1993), Bengio et al. (1994) and Siegelmann and Sontag (1995), with a review in Pascanu et al. (2013a). The remaining sections of this chapter discuss various approaches that have been proposed to reduce the diﬃculty of learning longterm dependencies (in some cases allowing an RNN to learn dependencies across hundreds of steps), but the problem of learning long-term dependencies remains one of the main challenges in deep learning.

10.8

Echo State Networks

The recurrent weights mapping from h(t−1) to h(t) and the input weights mapping from x(t) to h(t) are some of the most diﬃcult parameters to learn in a recurrent network. One proposed (Jaeger, 2003; Maass et al., 2002; Jaeger and Haas, 2004; Jaeger, 2007b) approach to avoiding this diﬃculty is to set the recurrent weights such that the recurrent hidden units do a good job of capturing the history of past inputs, and only learn the output weights. This is the idea that was independently proposed for echo state networks or ESNs (Jaeger and Haas, 2004; Jaeger, 2007b) and liquid state machines (Maass et al., 2002). The latter is similar, except that it uses spiking neurons (with binary outputs) instead of the continuousvalued hidden units used for ESNs. Both ESNs and liquid state machines are termed reservoir computing (Lukoševičius and Jaeger, 2009) to denote the fact that the hidden units form of reservoir of temporal features which may capture diﬀerent aspects of the history of inputs. One way to think about these reservoir computing recurrent networks is that they are similar to kernel machines: they map an arbitrary length sequence (the history of inputs up to time t) into a ﬁxed-length vector (the recurrent state h(t) ), on which a linear predictor (typically a linear regression) can be applied to solve the problem of interest. The training criterion may then be easily designed to be convex as a function of the output weights. For example, if the output consists of linear regression from the hidden units to the output targets, and the training criterion is mean squared error, then it is convex and may be solved reliably with simple learning algorithms (Jaeger, 2003). The important question is therefore: how do we set the input and recurrent weights so that a rich set of histories can be represented in the recurrent neural network state? The answer proposed in the reservoir computing literature is to 405

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view the recurrent net as a dynamical system, and set the input and recurrent weights such that the dynamical system is near the edge of stability. The original idea was to make the eigenvalues of the Jacobian of the state-tostate transition function be close to 1. As explained in Sec. 8.2.5, an important characteristic of a recurrent network is the eigenvalue spectrum of the Jacobians (t) J (t) = ∂s∂s(t−1) . Of particular importance is the spectral radius of J (t) , deﬁned to be the maximum of the absolute values of its eigenvalues. To understand the eﬀect of the spectral radius, consider the simple case of back-propagation with a Jacobian matrix J that does not change with t. This case happens, for example, when the network is purely linear. Suppose that J has an eigenvector v with corresponding eigenvalue λ. Consider what happens as we propagate a gradient vector backwards through time. If we begin with a gradient vector g, then after one step of back-propagation, we will have J g, and after n steps we will have J ng. Now consider what happens if we instead back-propagate a perturbed version of g. If we begin with g + δv, then after one step, we will have J (g + δv). After n steps, we will have J n(g + δv ). From this we can see that back-propagation starting from g and back-propagation starting from g + δv diverge by δJ nv after n steps of back-propagation. If v is chosen to be a unit eigenvector of J with eigenvalue λ , then multiplication by the Jacobian simply scales the diﬀerence at each step. The two executions of back-propagation are separated by a distance of δ|λ| n. When v corresponds to the largest value of |λ|, this perturbation achieves the widest possible separation of an initial perturbation of size δ . When |λ| > 1, the deviation size δ|λ|n grows exponentially large. When |λ| < 1, the deviation size becomes exponentially small. Of course, this example assumed that the Jacobian was the same at every time step, corresponding to a recurrent network with no nonlinearity. When a nonlinearity is present, the derivative of the nonlinearity will approach zero on many time steps, and help to prevent the explosion resulting from a large spectral radius. Indeed, the most recent work on echo state networks advocates using a spectral radius much larger than unity (Yildiz et al., 2012; Jaeger, 2012). Everything we have said about back-propagation via repeated matrix multiplication applies equally to forward propagation in a network with no nonlinearity, where the state h(t+1) = h(t)W . When a linear map W always shrinks h as measured by the L2 norm, then we say that the map is contractive. When the spectral radius is less than one, the mapping from h(t) to h(t+1) is contractive, so a small change becomes smaller after each time step. This necessarily makes the network forget information about the 406

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past when we use a ﬁnite level of precision (such as 32 bit integers) to store the state vector. The Jacobian matrix tells us how a small change of h(t) propagates one step forward, or equivalently, how the gradient on h(t+1) propagates one step backward, during back-propagation. Note that neither W nor J need to be symmetric (although they are square and real), so they can have complex-valued eigenvalues and eigenvectors, with imaginary components corresponding to potentially oscillatory behavior (if the same Jacobian was applied iteratively). Even though h(t) or a small variation of h (t) of interest in back-propagation are real-valued, they can be expressed in such a complex-valued basis. What matters is what happens to the magnitude (complex absolute value) of these possibly complex-valued basis coeﬃcients, when we multiply the matrix by the vector. An eigenvalue with magnitude greater than one corresponds to magniﬁcation (exponential growth, if applied iteratively) or shrinking (exponential decay, if applied iteratively). With a nonlinear map, the Jacobian is free to change at each step. The dynamics therefore become more complicated. However, it remains true that a small initial variation can turn into a large variation after several steps. One diﬀerence between the purely linear case and the nonlinear case is that the use of a squashing nonlinearity such as tanh can cause the recurrent dynamics to become bounded. Note that it is possible for back-propagation to retain unbounded dynamics even when forward propagation has bounded dynamics, for example, when a sequence of tanh units are all in the middle of their linear regime and are connected by weight matrices with spectral radius greater than 1. However, it is rare for all of the tanh units to simultaneously lie at their linear activation point. The strategy of echo state networks is simply to ﬁx the weights to have some spectral radius such as 3, where information is carried forward through time but does not explode due to the stabilizing eﬀect of saturating nonlinearities like tanh. More recently, it has been shown that the techniques used to set the weights in ESNs could be used to initialize the weights in a fully trainable recurrent network (with the hidden-to-hidden recurrent weights trained using back-propagation through time), helping to learn long-term dependencies (Sutskever, 2012; Sutskever et al., 2013). In this setting, an initial spectral radius of 1.2 performs well, combined with the sparse initialization scheme described in Sec. 8.4.

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10.9

Leaky Units and Other Strategies for Multiple Time Scales

One way to deal with long-term dependencies is to design a model that operates at multiple time scales, so that some parts of the model operate at ﬁne-grained time scales and can handle small details, while other parts operate at coarse time scales and transfer information from the distant past to the present more eﬃciently. Various strategies for building both ﬁne and coarse time scales are possible. These include the addition of skip connections across time, “leaky units” that integrate signals with diﬀerent time constants, and the removal of some of the connections used to model ﬁne-grained time scales.

10.9.1

Adding Skip Connections through Time

One way to obtain coarse time scales is to add direct connections from variables in the distant past to variables in the present. The idea of using such skip connections dates back to Lin et al. (1996) and follows from the idea of incorporating delays in feedforward neural networks (Lang and Hinton, 1988). In an ordinary recurrent network, a recurrent connection goes from a unit at time t to a unit at time t + 1. It is possible to construct recurrent networks with longer delays (Bengio, 1991). As we have seen in Sec. 8.2.5, gradients may vanish or explode exponentially with respect to the number of time steps. Lin et al. (1996) introduced recurrent connections with a time-delay of d to mitigate this problem. Gradients now diminish exponentially as a function of τd rather than τ . Since there are both delayed and single step connections, gradients may still explode exponentially in τ. This allows the learning algorithm to capture longer dependencies although not all long-term dependencies may be represented well in this way.

10.9.2

Leaky Units and a Spectrum of Diﬀerent Time Scales

Another way to obtain paths on which the product of derivatives is close to one is to have units with linear self-connections and a weight near one on these connections. When we accumulate a running average µ(t) of some value v (t) by applying the update µ (t) ← αµ(t−1) + (1 − α)v(t) the α parameter is an example of a linear selfconnection from µ(t−1) to µ(t) . When α is near one, the running average remembers information about the past for a long time, and when α is near zero, information about the past is rapidly discarded. Hidden units with linear self-connections can behave similarly to such running averages. Such hidden units are called leaky units. 408

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Skip connections through d time steps are a way of ensuring that a unit can always learn to be inﬂuenced by a value from d time steps earlier. The use of a linear self-connection with a weight near one is a diﬀerent way of ensuring that the unit can access values from the past. The linear self-connection approach allows this eﬀect to be adapted more smoothly and ﬂexibly by adjusting the real-valued α rather than by adjusting the integer-valued skip length. These ideas were proposed by Mozer (1992) and by El Hihi and Bengio (1996). Leaky units were also found to be useful in the context of echo state networks (Jaeger et al., 2007). There are two basic strategies for setting the time constants used by leaky units. One strategy is to manually ﬁx them to values that remain constant, for example by sampling their values from some distribution once at initialization time. Another strategy is to make the time constants free parameters and learn them. Having such leaky units at diﬀerent time scales appears to help with long-term dependencies (Mozer, 1992; Pascanu et al., 2013a).

10.9.3

Removing Connections

Another approach to handle long-term dependencies is the idea of organizing the state of the RNN at multiple time-scales (El Hihi and Bengio, 1996), with information ﬂowing more easily through long distances at the slower time scales. This idea diﬀers from the skip connections through time discussed earlier because it involves actively removing length-one connections and replacing them with longer connections. Units modiﬁed in such a way are forced to operate on a long time scale. Skip connections through time add edges. Units receiving such new connections may learn to operate on a long time scale but may also choose to focus on their other short-term connections. There are diﬀerent ways in which a group of recurrent units can be forced to operate at diﬀerent time scales. One option is to make the recurrent units leaky, but to have diﬀerent groups of units associated with diﬀerent ﬁxed time scales. This was the proposal in Mozer (1992) and has been successfully used in Pascanu et al. (2013a). Another option is to have explicit and discrete updates taking place at diﬀerent times, with a diﬀerent frequency for diﬀerent groups of units. This is the approach of El Hihi and Bengio (1996) and Koutnik et al. (2014). It worked well on a number of benchmark datasets.

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10.10

The Long Short-Term Memory and Other Gated RNNs

As of this writing, the most eﬀective sequence models used in practical applications are called gated RNNs. These include the long short-term memory and networks based on the gated recurrent unit. Like leaky units, gated RNNs are based on the idea of creating paths through time that have derivatives that neither vanish nor explode. Leaky units did this with connection weights that were either manually chosen constants or were parameters. Gated RNNs generalize this to connection weights that may change at each time step. Leaky units allow the network to accumulate information (such as evidence for a particular feature or category) over a long duration. However, once that information has been used, it might be useful for the neural network to forget the old state. For example, if a sequence is made of sub-sequences and we want a leaky unit to accumulate evidence inside each sub-subsequence, we need a mechanism to forget the old state by setting it to zero. Instead of manually deciding when to clear the state, we want the neural network to learn to decide when to do it. This is what gated RNNs do.

10.10.1

LSTM

The clever idea of introducing self-loops to produce paths where the gradient can ﬂow for long durations is a core contribution of the initial long short-term memory (LSTM) model (Hochreiter and Schmidhuber, 1997). A crucial addition has been to make the weight on this self-loop conditioned on the context, rather than ﬁxed (Gers et al., 2000). By making the weight of this self-loop gated (controlled by another hidden unit), the time scale of integration can be changed dynamically. In this case, we mean that even for an LSTM with ﬁxed parameters, the time scale of integration can change based on the input sequence, because the time constants are output by the model itself. The LSTM has been found extremely successful in many applications, such as unconstrained handwriting recognition (Graves et al., 2009), speech recognition (Graves et al., 2013; Graves and Jaitly, 2014), handwriting generation (Graves, 2013), machine translation (Sutskever et al., 2014), image captioning (Kiros et al., 2014b; Vinyals et al., 2014b; Xu et al., 2015) and parsing (Vinyals et al., 2014a). The LSTM block diagram is illustrated in Fig. 10.16. The corresponding forward propagation equations are given below, in the case of a shallow recurrent 410

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output

×

self-loop

+

× state

×

input

input gate

forget gate

output gate

Figure 10.16: Block diagram of the LSTM recurrent network “cell.” Cells are connected recurrently to each other, replacing the usual hidden units of ordinary recurrent networks. An input feature is computed with a regular artiﬁcial neuron unit. Its value can be accumulated into the state if the sigmoidal input gate allows it. The state unit has a linear self-loop whose weight is controlled by the forget gate. The output of the cell can be shut oﬀ by the output gate. All the gating units have a sigmoid nonlinearity, while the input unit can have any squashing nonlinearity. The state unit can also be used as an extra input to the gating units. The black square indicates a delay of a single time step.

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network architecture. Deeper architectures have also been successfully used (Graves et al., 2013; Pascanu et al., 2014a). Instead of a unit that simply applies an elementwise nonlinearity to the aﬃne transformation of inputs and recurrent units, LSTM recurrent networks have “LSTM cells” that have an internal recurrence (a self-loop), in addition to the outer recurrence of the RNN. Each cell has the same inputs and outputs as an ordinary recurrent network, but has more parameters and a system of gating units that controls the ﬂow of information. The most important (t) component is the state unit si that has a linear self-loop similar to the leaky units described in the previous section. However, here, the self-loop weight (or the associated time constant) is controlled by a forget gate unit fi(t) (for time step t and cell i), that sets this weight to a value between 0 and 1 via a sigmoid unit: f (t) f (t−1) fi(t) = σ bfi + Ui,j xj + Wi,j hj , (10.40) j

j

where x (t) is the current input vector and h(t) is the current hidden layer vector, containing the outputs of all the LSTM cells, and bf ,Uf , W f are respectively biases, input weights and recurrent weights for the forget gates. The LSTM cell internal state is thus updated as follows, but with a conditional self-loop weight fi(t): (t−1) (t) (t) (t−1) (t) (t) si = f i si + g i σ b i + Ui,j xj + Wi,j hj , (10.41) j

j

where b, U and W respectively denote the biases, input weights and recurrent weights into the LSTM cell. The external input gate unit g(i t) is computed similarly to the forget gate (with a sigmoid unit to obtain a gating value between 0 and 1), but with its own parameters: g (t) g (t−1) g(i t) = σ bgi + Ui,j xj + Wi,j hj . (10.42) j

j

(t)

(t)

The output hi of the LSTM cell can also be shut oﬀ, via the output gate q i , which also uses a sigmoid unit for gating: (t) (t) (t) h i = tanh si qi (10.43) (t) o o (t) o (t−1) qi = σ bi + U i,j xj + Wi,j hj (10.44) j

j

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which has parameters bo , U o , W o for its biases, input weights and recurrent weights, respectively. Among the variants, one can choose to use the cell state s (i t) as an extra input (with its weight) into the three gates of the i-th unit, as shown in Fig. 10.16. This would require three additional parameters. LSTM networks have been shown to learn long-term dependencies more easily than the simple recurrent architectures, ﬁrst on artiﬁcial data sets designed for testing the ability to learn long-term dependencies (Bengio et al., 1994; Hochreiter and Schmidhuber, 1997; Hochreiter et al., 2001), then on challenging sequence processing tasks where state-of-the-art performance was obtained (Graves, 2012; Graves et al., 2013; Sutskever et al., 2014). Variants and alternatives to the LSTM have been studied and used and are discussed next.

10.10.2

Other Gated RNNs

Which pieces of the LSTM architecture are actually necessary? What other successful architectures could be designed that allow the network to dynamically control the time scale and forgetting behavior of diﬀerent units? Some answers to these questions are given with the recent work on gated RNNs, whose units are also known as gated recurrent units or GRUs (Cho et al., 2014b; Chung et al., 2014, 2015a; Jozefowicz et al., 2015; Chrupala et al., 2015). The main diﬀerence with the LSTM is that a single gating unit simultaneously controls the forgetting factor and the decision to update the state unit. The update equations are the following: (t−1) (t−1) (t−1) (t−1) (t−1) (t) + (1 − u (i t−1))σ bi + hi = ui hi Ui,j x j + Wi,j rj hj , j

j

(10.45) where u stands for “update” gate and r for “reset” gate. Their value is deﬁned as usual: (t) u (t) u (t) ui = σ bui + Ui,j xj + Wi,j hj (10.46) j

and

r (i t) = σ b ri +

j

r (t) U i,j xj +

j

j

r (t) Wi,j hj .

(10.47)

The reset and updates gates can individually “ignore” parts of the state vector. The update gates act like conditional leaky integrators that can linearly gate any 413

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dimension, thus choosing to copy it (at one extreme of the sigmoid) or completely ignore it (at the other extreme) by replacing it by the new “target state” value (towards which the leaky integrator wants to converge). The reset gates control which parts of the state get used to compute the next target state, introducing an additional nonlinear eﬀect in the relationship between past state and future state. Many more variants around this theme can be designed. For example the reset gate (or forget gate) output could be shared across multiple hidden units. Alternately, the product of a global gate (covering a whole group of units, such as an entire layer) and a local gate (per unit) could be used to combine global control and local control. However, several investigations over architectural variations of the LSTM and GRU found no variant that would clearly beat both of these across a wide range of tasks (Greﬀ et al., 2015; Jozefowicz et al., 2015). Greﬀ et al. (2015) found that a crucial ingredient is the forget gate, while Jozefowicz et al. (2015) found that adding a bias of 1 to the LSTM forget gate, a practice advocated by Gers et al. (2000), makes the LSTM as strong as the best of the explored architectural variants.

10.11

Optimization for Long-Term Dependencies

Sec. 8.2.5 and Sec. 10.7 have described the vanishing and exploding gradient problems that occur when optimizing RNNs over many time steps. An interesting idea proposed by Martens and Sutskever (2011) is that second derivatives may vanish at the same time that ﬁrst derivatives vanish. Second-order optimization algorithms may roughly be understood as dividing the ﬁrst derivative by the second derivative (in higher dimension, multiplying the gradient by the inverse Hessian). If the second derivative shrinks at a similar rate to the ﬁrst derivative, then the ratio of ﬁrst and second derivatives may remain relatively constant. Unfortunately, second-order methods have many drawbacks, including high computational cost, the need for a large minibatch, and a tendency to be attracted to saddle points. Martens and Sutskever (2011) found promising results using second-order methods. Later, Sutskever et al. (2013) found that simpler methods such as Nesterov momentum with careful initialization could achieve similar results. See Sutskever (2012) for more detail. Both of these approaches have largely been replaced by simply using SGD (even without momentum) applied to LSTMs. This is part of a continuing theme in machine learning that it is often much easier to design a model that is easy to optimize than it is to design a more powerful optimization algorithm. 414

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10.11.1

Clipping Gradients

As discussed in Sec. 8.2.4, strongly nonlinear functions such as those computed by a recurrent net over many time steps tend to have derivatives that can be either very large or very small in magnitude. This is illustrated in Fig. 8.3 and Fig. 10.17, in which we see that the objective function (as a function of the parameters) has a “landscape” in which one ﬁnds “cliﬀs”: wide and rather ﬂat regions separated by tiny regions where the objective function changes quickly, forming a kind of cliﬀ. The diﬃculty that arises is that when the parameter gradient is very large, a gradient descent parameter update could throw the parameters very far, into a region where the objective function is larger, undoing much of the work that had been done to reach the current solution. The gradient tells us the direction that corresponds to the steepest descent within an inﬁnitesimal region surrounding the current parameters. Outside of this inﬁnitesimal region, the cost function may begin to curve back upwards. The update must be chosen to be small enough to avoid traversing too much upward curvature. We typically use learning rates that decay slowly enough that consecutive steps have approximately the same learning rate. A step size that is appropriate for a relatively linear part of the landscape is often inappropriate and causes uphill motion if we enter a more curved part of the landscape on the next step.

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Figure 10.17: Example of the eﬀect of gradient clipping in a recurrent network with two parameters w and b . Gradient clipping can make gradient descent perform more reasonably in the vicinity of extremely steep cliﬀs. These steep cliﬀs commonly occur in recurrent networks near where a recurrent network behaves approximately linearly. The cliﬀ is exponentially steep in the number of time steps because the weight matrix is multiplied by itself once for each time step. (Left) Gradient descent without gradient clipping overshoots the bottom of this small ravine, then receives a very large gradient from the cliﬀ face. The large gradient catastrophically propels the parameters outside the axes of the plot. (Right) Gradient descent with gradient clipping has a more moderate reaction to the cliﬀ. While it does ascend the cliﬀ face, the step size is restricted so that it cannot be propelled away from steep region near the solution. Figure adapted with permission from Pascanu et al. (2013a).

A simple type of solution has been in use by practitioners for many years: clipping the gradient. There are diﬀerent instances of this idea (Mikolov, 2012; Pascanu et al., 2013a). One option is to clip the parameter gradient from a minibatch element-wise (Mikolov, 2012) just before the parameter update. Another is to clip the norm ||g|| of the gradient g (Pascanu et al., 2013a) just before the parameter update: if ||g|| > v g←

(10.48) gv ||g||

(10.49)

where v is the norm threshold and g is used to update parameters. Because the gradient of all the parameters (including diﬀerent groups of parameters, such as weights and biases) is renormalized jointly with a single scaling factor, the latter method has the advantage that it guarantees that each step is still in the gradient direction, but experiments suggest that both forms work similarly. Although 416

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the parameter update has the same direction as the true gradient, with gradient norm clipping, the parameter update vector norm is now bounded. This bounded gradient avoids performing a detrimental step when the gradient explodes. In fact, even simply taking a random step when the gradient magnitude is above a threshold tends to work almost as well. If the explosion is so severe that the gradient is numerically Inf or Nan (considered inﬁnite or not-a-number), then a random step of size v can be taken and will typically move away from the numerically unstable conﬁguration. Clipping the gradient norm per-minibatch will not change the direction of the gradient for an individual minibatch. However, taking the average of the norm-clipped gradient from many minibatches is not equivalent to clipping the norm of the true gradient (the gradient formed from using all examples). Examples that have large gradient norm, as well as examples that appear in the same minibatch as such examples, will have their contribution to the ﬁnal direction diminished. This stands in contrast to traditional minibatch gradient descent, where the true gradient direction is equal to the average over all minibatch gradients. Put another way, traditional stochastic gradient descent uses an unbiased estimate of the gradient, while gradient descent with norm clipping introduces a heuristic bias that we know empirically to be useful. With elementwise clipping, the direction of the update is not aligned with the true gradient or the minibatch gradient, but it is still a descent direction. It has also been proposed (Graves, 2013) to clip the back-propagated gradient (with respect to hidden units) but no comparison has been published between these variants; we conjecture that all these methods behave similarly.

10.11.2

Regularizing to Encourage Information Flow

Gradient clipping helps to deal with exploding gradients, but it does not help with vanishing gradients. To address vanishing gradients and better capture long-term dependencies, we discussed the idea of creating paths in the computational graph of the unfolded recurrent architecture along which the product of gradients associated with arcs is near 1. One approach to achieve this is with LSTMs and other self-loops and gating mechanisms, described above in Sec. 10.10. Another idea is to regularize or constrain the parameters so as to encourage “information ﬂow.” In particular, we would like the gradient vector ∇ h(t) L being back-propagated to maintain its magnitude, even if the loss function only penalizes the output at the end of the sequence. Formally, we want (∇h(t) L)

∂h (t) ∂h(t−1)

417

(10.50)

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to be as large as ∇h(t) L.

(10.51)

With this objective, Pascanu et al. (2013a) propose the following regularizer: 2 ∂h(t) | L) | ( ∇ h(t) ∂h(t−1) − 1 . Ω= ||∇ L| | ( t ) h t

(10.52)

Computing the gradient of this regularizer may appear diﬃcult, but Pascanu et al. (2013a) propose an approximation in which we consider the back-propagated vectors ∇h (t) L as if they were constants (for the purpose of this regularizer, so that there is no need to back-propagate through them). The experiments with this regularizer suggest that, if combined with the norm clipping heuristic (which handles gradient explosion), the regularizer can considerably increase the span of the dependencies that an RNN can learn. Because it keeps the RNN dynamics on the edge of explosive gradients, the gradient clipping is particularly important. Without gradient clipping, gradient explosion prevents learning from succeeding. A key weakness of this approach is that it is not as eﬀective as the LSTM for tasks where data is abundant, such as language modeling.

10.12

Explicit Memory

Intelligence requires knowledge and acquiring knowledge can be done via learning, which has motivated the development of large-scale deep architectures. However, there are diﬀerent kinds of knowledge. Some knowledge can be implicit, subconscious, and diﬃcult to verbalize—such as how to walk, or how a dog looks diﬀerent from a cat. Other knowledge can be explicit, declarative, and relatively straightforward to put into words—every day commonsense knowledge, like “a cat is a kind of animal,” or very speciﬁc facts that you need to know to accomplish your current goals, like “the meeting with the sales team is at 3:00 PM in room 141.” Neural networks excel at storing implicit knowledge. However, they struggle to memorize facts. Stochastic gradient descent requires many presentations of the same input before it can be stored in a neural network parameters, and even then, that input will not be stored especially precisely. Graves et al. (2014b) hypothesized that this is because neural networks lack the equivalent of the working memory system that allows human beings to explicitly hold and manipulate pieces of information that are relevant to achieving some goal. Such explicit memory 418

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Memory cells

Reading mechanism

Writing mechanism

Task network, controlling the memory

Figure 10.18: A schematic of an example of a network with an explicit memory, capturing some of the key design elements of the neural Turing machine. In this diagram we distinguish the “representation” part of the model (the “task network,” here a recurrent net in the bottom) from the “memory” part of the model (the set of cells), which can store facts. The task network learns to “control” the memory, deciding where to read from and where to write to within the memory (through the reading and writing mechanisms, indicated by bold arrows pointing at the reading and writing addresses).

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components would allow our systems not only to rapidly and “intentionally” store and retrieve speciﬁc facts but also to sequentially reason with them. The need for neural networks that can process information in a sequence of steps, changing the way the input is fed into the network at each step, has long been recognized as important for the ability to reason rather than to make automatic, intuitive responses to the input (Hinton, 1990). To resolve this diﬃculty, Weston et al. (2014) introduced memory networks that include a set of memory cells that can be accessed via an addressing mechanism. Memory networks originally required a supervision signal instructing them how to use their memory cells. Graves et al. (2014b) introduced the neural Turing machine, which is able to learn to read from and write arbitrary content to memory cells without explicit supervision about which actions to undertake, and allowed end-to-end training without this supervision signal, via the use of a content-based soft attention mechanism (see Bahdanau et al. (2015) and Sec. 12.4.5.1). This soft addressing mechanism has become standard with other related architectures emulating algorithmic mechanisms in a way that still allows gradient-based optimization (Sukhbaatar et al., 2015; Joulin and Mikolov, 2015; Kumar et al., 2015; Vinyals et al., 2015a; Grefenstette et al., 2015). Each memory cell can be thought of as an extension of the memory cells in LSTMs and GRUs. The diﬀerence is that the network outputs an internal state that chooses which cell to read from or write to, just as memory accesses in a digital computer read from or write to a speciﬁc address. It is diﬃcult to optimize functions that produce exact, integer addresses. To alleviate this problem, NTMs actually read to or write from many memory cells simultaneously. To read, they take a weighted average of many cells. To write, they modify multiple cells by diﬀerent amounts. The coeﬃcients for these operations are chosen to be focused on a small number of cells, for example, by producing them via a softmax function. Using these weights with non-zero derivatives allows the functions controlling access to the memory to be optimized using gradient descent. The gradient on these coeﬃcients indicates whether each of them should be increased or decreased, but the gradient will typically be large only for those memory addresses receiving a large coeﬃcient. These memory cells are typically augmented to contain a vector, rather than the single scalar stored by an LSTM or GRU memory cell. There are two reasons to increase the size of the memory cell. One reason is that we have increased the cost of accessing a memory cell. We pay the computational cost of producing a coeﬃcient for many cells, but we expect these coeﬃcients to cluster around a small number of cells. By reading a vector value, rather than a scalar value, we can 420

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oﬀset some of this cost. Another reason to use vector-valued memory cells is that they allow for content-based addressing, where the weight used to read to or write from a cell is a function of that cell. Vector-valued cells allow us to retrieve a complete vector-valued memory if we are able to produce a pattern that matches some but not all of its elements. This is analogous to the way that people can recall the lyrics of a song based on a few words. We can think of a content-based read instruction as saying, “Retrieve the lyrics of the song that has the chorus ‘We all live in a yellow submarine.’ ” Content-based addressing is more useful when we make the objects to be retrieved large—if every letter of the song was stored in a separate memory cell, we would not be able to ﬁnd them this way. By comparison, location-based addressing is not allowed to refer to the content of the memory. We can think of a location-based read instruction as saying “Retrieve the lyrics of the song in slot 347.” Location-based addressing can often be a perfectly sensible mechanism even when the memory cells are small. If the content of a memory cell is copied (not forgotten) at most time steps, then the information it contains can be propagated forward in time and the gradients propagated backward in time without either vanishing or exploding. The explicit memory approach is illustrated in Fig. 10.18, where we see that a “task neural network” is coupled with a memory. Although that task neural network could be feedforward or recurrent, the overall system is a recurrent network. The task network can choose to read from or write to speciﬁc memory addresses. Explicit memory seems to allow models to learn tasks that ordinary RNNs or LSTM RNNs cannot learn. One reason for this advantage may be because information and gradients can be propagated (forward in time or backwards in time, respectively) for very long durations. As an alternative to back-propagation through weighted averages of memory cells, we can interpret the memory addressing coeﬃcients as probabilities and stochastically read just one cell (Zaremba and Sutskever, 2015). Optimizing models that make discrete decisions requires specialized optimization algorithms, described in Sec. 20.9.1. So far, training these stochastic architectures that make discrete decisions remains harder than training deterministic algorithms that make soft decisions. Whether it is soft (allowing back-propagation) or stochastic and hard, the mechanism for choosing an address is in its form identical to the attention mechanism which had been previously introduced in the context of machine translation (Bahdanau et al., 2015) and discussed in Sec. 12.4.5.1. The idea of attention mechanisms for neural networks was introduced even earlier, in the context of handwriting generation (Graves, 2013), with an attention mechanism that was constrained to 421

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move only forward in time through the sequence. In the case of machine translation and memory networks, at each step, the focus of attention can move to a completely diﬀerent place, compared to the previous step. Recurrent neural networks provide a way to extend deep learning to sequential data. They are the last major tool in our deep learning toolbox. Our discussion now moves to how to choose and use these tools and how to apply them to real-world tasks.

422

Chapter 11

Practical Methodology Successfully applying deep learning techniques requires more than just a good knowledge of what algorithms exist and the principles that explain how they work. A good machine learning practitioner also needs to know how to choose an algorithm for a particular application and how to monitor and respond to feedback obtained from experiments in order to improve a machine learning system. During day to day development of machine learning systems, practitioners need to decide whether to gather more data, increase or decrease model capacity, add or remove regularizing features, improve the optimization of a model, improve approximate inference in a model, or debug the software implementation of the model. All of these operations are at the very least time-consuming to try out, so it is important to be able to determine the right course of action rather than blindly guessing. Most of this book is about diﬀerent machine learning models, training algorithms, and objective functions. This may give the impression that the most important ingredient to being a machine learning expert is knowing a wide variety of machine learning techniques and being good at diﬀerent kinds of math. In practice, one can usually do much better with a correct application of a commonplace algorithm than by sloppily applying an obscure algorithm. Correct application of an algorithm depends on mastering some fairly simple methodology. Many of the recommendations in this chapter are adapted from Ng (2015). We recommend the following practical design process: • Determine your goals—what error metric to use, and your target value for this error metric. These goals and error metrics should be driven by the problem that the application is intended to solve. • Establish a working end-to-end pipeline as soon as possible, including the 423

CHAPTER 11. PRACTICAL METHODOLOGY

estimation of the appropriate performance metrics. • Instrument the system well to determine bottlenecks in performance. Diagnose which components are performing worse than expected and whether it is due to overﬁtting, underﬁtting, or a defect in the data or software. • Repeatedly make incremental changes such as gathering new data, adjusting hyperparameters, or changing algorithms, based on speciﬁc ﬁndings from your instrumentation. As a running example, we will use Street View address number transcription system (Goodfellow et al., 2014d). The purpose of this application is to add buildings to Google Maps. Street View cars photograph the buildings and record the GPS coordinates associated with each photograph. A convolutional network recognizes the address number in each photograph, allowing the Google Maps database to add that address in the correct location. The story of how this commercial application was developed gives an example of how to follow the design methodology we advocate. We now describe each of the steps in this process.

11.1

Performance Metrics

Determining your goals, in terms of which error metric to use, is a necessary ﬁrst step because your error metric will guide all of your future actions. You should also have an idea of what level of performance you desire. Keep in mind that for most applications, it is impossible to achieve absolute zero error. The Bayes error deﬁnes the minimum error rate that you can hope to achieve, even if you have inﬁnite training data and can recover the true probability distribution. This is because your input features may not contain complete information about the output variable, or because the system might be intrinsically stochastic. You will also be limited by having a ﬁnite amount of training data. The amount of training data can be limited for a variety of reasons. When your goal is to build the best possible real-world product or service, you can typically collect more data but must determine the value of reducing error further and weigh this against the cost of collecting more data. Data collection can require time, money, or human suﬀering (for example, if your data collection process involves performing invasive medical tests). When your goal is to answer a scientiﬁc question about which algorithm performs better on a ﬁxed benchmark, the benchmark 424

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speciﬁcation usually determines the training set and you are not allowed to collect more data. How can one determine a reasonable level of performance to expect? Typically, in the academic setting, we have some estimate of the error rate that is attainable based on previously published benchmark results. In the real-word setting, we have some idea of the error rate that is necessary for an application to be safe, cost-eﬀective, or appealing to consumers. Once you have determined your realistic desired error rate, your design decisions will be guided by reaching this error rate. Another important consideration besides the target value of the performance metric is the choice of which metric to use. Several diﬀerent performance metrics may be used to measure the eﬀectiveness of a complete application that includes machine learning components. These performance metrics are usually diﬀerent from the cost function used to train the model. As described in Sec. 5.1.2, it is common to measure the accuracy, or equivalently, the error rate, of a system. However, many applications require more advanced metrics. Sometimes it is much more costly to make one kind of a mistake than another. For example, an e-mail spam detection system can make two kinds of mistakes: incorrectly classifying a legitimate message as spam, and incorrectly allowing a spam message to appear in the inbox. It is much worse to block a legitimate message than to allow a questionable message to pass through. Rather than measuring the error rate of a spam classiﬁer, we may wish to measure some form of total cost, where the cost of blocking legitimate messages is higher than the cost of allowing spam messages. Sometimes we wish to train a binary classiﬁer that is intended to detect some rare event. For example, we might design a medical test for a rare disease. Suppose that only one in every million people has this disease. We can easily achieve 99.9999% accuracy on the detection task, by simply hard-coding the classiﬁer to always report that the disease is absent. Clearly, accuracy is a poor way to characterize the performance of such a system. One way to solve this problem is to instead measure precision and recall. Precision is the fraction of detections reported by the model that were correct, while recall is the fraction of true events that were detected. A detector that says no one has the disease would achieve perfect precision, but zero recall. A detector that says everyone has the disease would achieve perfect recall, but precision equal to the percentage of people who have the disease (0.0001% in our example of a disease that only one people in a million have). When using precision and recall, it is common to plot a PR curve, with precision on the y-axis and recall on the x-axis. The classiﬁer generates a score that is higher if the event to be detected occurred. For example, a feedforward 425

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network designed to detect a disease outputs yˆ = P (y = 1 | x), estimating the probability that a person whose medical results are described by features x has the disease. We choose to report a detection whenever this score exceeds some threshold. By varying the threshold, we can trade precision for recall. In many cases, we wish to summarize the performance of the classiﬁer with a single number rather than a curve. To do so, we can convert precision p and recall r into an F-score given by 2pr F = . (11.1) p+r Another option is to report the total area lying beneath the PR curve. In some applications, it is possible for the machine learning system to refuse to make a decision. This is useful when the machine learning algorithm can estimate how conﬁdent it should be about a decision, especially if a wrong decision can be harmful and if a human operator is able to occasionally take over. The Street View transcription system provides an example of this situation. The task is to transcribe the address number from a photograph in order to associate the location where the photo was taken with the correct address in a map. Because the value of the map degrades considerably if the map is inaccurate, it is important to add an address only if the transcription is correct. If the machine learning system thinks that it is less likely than a human being to obtain the correct transcription, then the best course of action is to allow a human to transcribe the photo instead. Of course, the machine learning system is only useful if it is able to dramatically reduce the amount of photos that the human operators must process. A natural performance metric to use in this situation is coverage. Coverage is the fraction of examples for which the machine learning system is able to produce a response. It is possible to trade coverage for accuracy. One can always obtain 100% accuracy by refusing to process any example, but this reduces the coverage to 0%. For the Street View task, the goal for the project was to reach human-level transcription accuracy while maintaining 95% coverage. Human-level performance on this task is 98% accuracy. Many other metrics are possible. We can for example, measure click-through rates, collect user satisfaction surveys, and so on. Many specialized application areas have application-speciﬁc criteria as well. What is important is to determine which performance metric to improve ahead of time, then concentrate on improving this metric. Without clearly deﬁned goals, it can be diﬃcult to tell whether changes to a machine learning system make progress or not.

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11.2

Default Baseline Models

After choosing performance metrics and goals, the next step in any practical application is to establish a reasonable end-to-end system as soon as possible. In this section, we provide recommendations for which algorithms to use as the ﬁrst baseline approach in various situations. Keep in mind that deep learning research progresses quickly, so better default algorithms are likely to become available soon after this writing. Depending on the complexity of your problem, you may even want to begin without using deep learning. If your problem has a chance of being solved by just choosing a few linear weights correctly, you may want to begin with a simple statistical model like logistic regression. If you know that your problem falls into an “AI-complete” category like object recognition, speech recognition, machine translation, and so on, then you are likely to do well by beginning with an appropriate deep learning model. First, choose the general category of model based on the structure of your data. If you want to perform supervised learning with ﬁxed-size vectors as input, use a feedforward network with fully connected layers. If the input has known topological structure (for example, if the input is an image), use a convolutional network. In these cases, you should begin by using some kind of piecewise linear unit (ReLUs or their generalizations like Leaky ReLUs, PreLus and maxout). If your input or output is a sequence, use a gated recurrent net (LSTM or GRU). A reasonable choice of optimization algorithm is SGD with momentum with a decaying learning rate (popular decay schemes that perform better or worse on diﬀerent problems include decaying linearly until reaching a ﬁxed minimum learning rate, decaying exponentially, or decreasing the learning rate by a factor of 2-10 each time validation error plateaus). Another very reasonable alternative is Adam. Batch normalization can have a dramatic eﬀect on optimization performance, especially for convolutional networks and networks with sigmoidal nonlinearities. While it is reasonable to omit batch normalization from the very ﬁrst baseline, it should be introduced quickly if optimization appears to be problematic. Unless your training set contains tens of millions of examples or more, you should include some mild forms of regularization from the start. Early stopping should be used almost universally. Dropout is an excellent regularizer that is easy to implement and compatible with many models and training algorithms. Batch normalization also sometimes reduces generalization error and allows dropout to be omitted, due to the noise in the estimate of the statistics used to normalize each variable. 427

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If your task is similar to another task that has been studied extensively, you will probably do well by ﬁrst copying the model and algorithm that is already known to perform best on the previously studied task. You may even want to copy a trained model from that task. For example, it is common to use the features from a convolutional network trained on ImageNet to solve other computer vision tasks (Girshick et al., 2015). A common question is whether to begin by using unsupervised learning, described further in Part III. This is somewhat domain speciﬁc. Some domains, such as natural language processing, are known to beneﬁt tremendously from unsupervised learning techniques such as learning unsupervised word embeddings. In other domains, such as computer vision, current unsupervised learning techniques do not bring a beneﬁt, except in the semi-supervised setting, when the number of labeled examples is very small (Kingma et al., 2014; Rasmus et al., 2015). If your application is in a context where unsupervised learning is known to be important, then include it in your ﬁrst end-to-end baseline. Otherwise, only use unsupervised learning in your ﬁrst attempt if the task you want to solve is unsupervised. You can always try adding unsupervised learning later if you observe that your initial baseline overﬁts.

11.3

Determining Whether to Gather More Data

After the ﬁrst end-to-end system is established, it is time to measure the performance of the algorithm and determine how to improve it. Many machine learning novices are tempted to make improvements by trying out many diﬀerent algorithms. However, it is often much better to gather more data than to improve the learning algorithm. How does one decide whether to gather more data? First, determine whether the performance on the training set is acceptable. If performance on the training set is poor, the learning algorithm is not using the training data that is already available, so there is no reason to gather more data. Instead, try increasing the size of the model by adding more layers or adding more hidden units to each layer. Also, try improving the learning algorithm, for example by tuning the learning rate hyperparameter. If large models and carefully tuned optimization algorithms do not work well, then the problem might be the quality of the training data. The data may be too noisy or may not include the right inputs needed to predict the desired outputs. This suggests starting over, collecting cleaner data or collecting a richer set of features. If the performance on the training set is acceptable, then measure the per428

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formance on a test set. If the performance on the test set is also acceptable, then there is nothing left to be done. If test set performance is much worse than training set performance, then gathering more data is one of the most eﬀective solutions. The key considerations are the cost and feasibility of gathering more data, the cost and feasibility of reducing the test error by other means, and the amount of data that is expected to be necessary to improve test set performance signiﬁcantly. At large internet companies with millions or billions of users, it is feasible to gather large datasets, and the expense of doing so can be considerably less than the other alternatives, so the answer is almost always to gather more training data. For example, the development of large labeled datasets was one of the most important factors in solving object recognition. In other contexts, such as medical applications, it may be costly or infeasible to gather more data. A simple alternative to gathering more data is to reduce the size of the model or improve regularization, by adjusting hyperparameters such as weight decay coeﬃcients, or by adding regularization strategies such as dropout. If you ﬁnd that the gap between train and test performance is still unacceptable even after tuning the regularization hyperparameters, then gathering more data is advisable. When deciding whether to gather more data, it is also necessary to decide how much to gather. It is helpful to plot curves showing the relationship between training set size and generalization error, like in Fig. 5.4. By extrapolating such curves, one can predict how much additional training data would be needed to achieve a certain level of performance. Usually, adding a small fraction of the total number of examples will not have a noticeable impact on generalization error. It is therefore recommended to experiment with training set sizes on a logarithmic scale, for example doubling the number of examples between consecutive experiments. If gathering much more data is not feasible, the only other way to improve generalization error is to improve the learning algorithm itself. This becomes the domain of research and not the domain of advice for applied practitioners.

11.4

Selecting Hyperparameters

Most deep learning algorithms come with many hyperparameters that control many aspects of the algorithm’s behavior. Some of these hyperparameters aﬀect the time and memory cost of running the algorithm. Some of these hyperparameters aﬀect the quality of the model recovered by the training process and its ability to infer correct results when deployed on new inputs. There are two basic approaches to choosing these hyperparameters: choosing them manually and choosing them automatically. Choosing the hyperparameters 429

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manually requires understanding what the hyperparameters do and how machine learning models achieve good generalization. Automatic hyperparameter selection algorithms greatly reduce the need to understand these ideas, but they are often much more computationally costly.

11.4.1

Manual Hyperparameter Tuning

To set hyperparameters manually, one must understand the relationship between hyperparameters, training error, generalization error and computational resources (memory and runtime). This means establishing a solid foundation on the fundamental ideas concerning the eﬀective capacity of a learning algorithm from Chapter 5. The goal of manual hyperparameter search is usually to ﬁnd the lowest generalization error subject to some runtime and memory budget. We do not discuss how to determine the runtime and memory impact of various hyperparameters here because this is highly platform-dependent. The primary goal of manual hyperparameter search is to adjust the eﬀective capacity of the model to match the complexity of the task. Eﬀective capacity is constrained by three factors: the representational capacity of the model, the ability of the learning algorithm to successfully minimize the cost function used to train the model, and the degree to which the cost function and training procedure regularize the model. A model with more layers and more hidden units per layer has higher representational capacity—it is capable of representing more complicated functions. It can not necessarily actually learn all of these functions though, if the training algorithm cannot discover that certain functions do a good job of minimizing the training cost, or if regularization terms such as weight decay forbid some of these functions. The generalization error typically follows a U-shaped curve when plotted as a function of one of the hyperparameters, as in Fig. 5.3. At one extreme, the hyperparameter value corresponds to low capacity, and generalization error is high because training error is high. This is the underﬁtting regime. At the other extreme, the hyperparameter value corresponds to high capacity, and the generalization error is high because the gap between training and test error is high. Somewhere in the middle lies the optimal model capacity, which achieves the lowest possible generalization error, by adding a medium generalization gap to a medium amount of training error. For some hyperparameters, overﬁtting occurs when the value of the hyperparameter is large. The number of hidden units in a layer is one such example, 430

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because increasing the number of hidden units increases the capacity of the model. For some hyperparameters, overﬁtting occurs when the value of the hyperparameter is small. For example, the smallest allowable weight decay coeﬃcient of zero corresponds to the greatest eﬀective capacity of the learning algorithm. Not every hyperparameter will be able to explore the entire U-shaped curve. Many hyperparameters are discrete, such as the number of units in a layer or the number of linear pieces in a maxout unit, so it is only possible to visit a few points along the curve. Some hyperparameters are binary. Usually these hyperparameters are switches that specify whether or not to use some optional component of the learning algorithm, such as a preprocessing step that normalizes the input features by subtracting their mean and dividing by their standard deviation. These hyperparameters can only explore two points on the curve. Other hyperparameters have some minimum or maximum value that prevents them from exploring some part of the curve. For example, the minimum weight decay coeﬃcient is zero. This means that if the model is underﬁtting when weight decay is zero, we can not enter the overﬁtting region by modifying the weight decay coeﬃcient. In other words, some hyperparameters can only subtract capacity. The learning rate is perhaps the most important hyperparameter. If you have time to tune only one hyperparameter, tune the learning rate. It controls the eﬀective capacity of the model in a more complicated way than other hyperparameters—the eﬀective capacity of the model is highest when the learning rate is correct for the optimization problem, not when the learning rate is especially large or especially small. The learning rate has a U-shaped curve for training error, illustrated in Fig. 11.1. When the learning rate is too large, gradient descent can inadvertently increase rather than decrease the training error. In the idealized quadratic case, this occurs if the learning rate is at least twice as large as its optimal value (LeCun et al., 1998a). When the learning rate is too small, training is not only slower, but may become permanently stuck with a high training error. This eﬀect is poorly understood (it would not happen for a convex loss function). Tuning the parameters other than the learning rate requires monitoring both training and test error to diagnose whether your model is overﬁtting or underﬁtting, then adjusting its capacity appropriately. If your error on the training set is higher than your target error rate, you have no choice but to increase capacity. If you are not using regularization and you are conﬁdent that your optimization algorithm is performing correctly, then you must add more layers to your network or add more hidden units. Unfortunately, this increases the computational costs associated with the model. If your error on the test set is higher than than your target error rate, you can 431

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8

Training error

7 6 5 4 3 2 1 0 10−2

10−1

10 0

Learning rate (logarithmic scale)

Figure 11.1: Typical relationship between the learning rate and the training error. Notice the sharp rise in error when the learning is above an optimal value. This is for a ﬁxed training time, as a smaller learning rate may sometimes only slow down training by a factor proportional to the learning rate reduction. Generalization error can follow this curve or be complicated by regularization eﬀects arising out of having a too large or too small learning rates, since poor optimization can, to some degree, reduce or prevent overﬁtting, and even points with equivalent training error can have diﬀerent generalization error.

now take two kinds of actions. The test error is the sum of the training error and the gap between training and test error. The optimal test error is found by trading oﬀ these quantities. Neural networks typically perform best when the training error is very low (and thus, when capacity is high) and the test error is primarily driven by the gap between train and test error. Your goal is to reduce this gap without increasing training error faster than the gap decreases. To reduce the gap, change regularization hyperparameters to reduce eﬀective model capacity, such as by adding dropout or weight decay. Usually the best performance comes from a large model that is regularized well, for example by using dropout. Most hyperparameters can be set by reasoning about whether they increase or decrease model capacity. Some examples are included in Table 11.1. While manually tuning hyperparameters, do not lose sight of your end goal: good performance on the test set. Adding regularization is only one way to achieve this goal. As long as you have low training error, you can always reduce generalization error by collecting more training data. The brute force way to practically guarantee success is to continually increase model capacity and training set size until the task is solved. This approach does of course increase the computational cost of training and inference, so it is only feasible given appropriate resources. In 432

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Hyperparameter

Increases capacity when. . . increased

Reason

Caveats

Increasing the number of hidden units increases the representational capacity of the model.

Increasing the number of hidden units increases both the time and memory cost of essentially every operation on the model.

Learning rate

tuned optimally

Convolution kernel width

increased

An improper learning rate, whether too high or too low, results in a model with low eﬀective capacity due to optimization failure Increasing the kernel width increases the number of parameters in the model

Implicit padding

zero

increased

Weight decay coeﬃcient

decreased

Dropout rate

decreased

Number of hidden units

Adding implicit zeros before convolution keeps the representation size large Decreasing the weight decay coeﬃcient frees the model parameters to become larger Dropping units less often gives the units more opportunities to “conspire” with each other to ﬁt the training set

A wider kernel results in a narrower output dimension, reducing model capacity unless you use implicit zero padding to reduce this eﬀect. Wider kernels require more memory for parameter storage and increase runtime, but a narrower output reduces memory cost. Increased time and memory cost of most operations.

Table 11.1: The eﬀect of various hyperparameters on model capacity.

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principle, this approach could fail due to optimization diﬃculties, but for many problems optimization does not seem to be a signiﬁcant barrier, provided that the model is chosen appropriately.

11.4.2

Automatic Hyperparameter Optimization Algorithms

The ideal learning algorithm just takes a dataset and outputs a function, without requiring hand-tuning of hyperparameters. The popularity of several learning algorithms such as logistic regression and SVMs stems in part from their ability to perform well with only one or two tuned hyperparameters. Neural networks can sometimes perform well with only a small number of tuned hyperparameters, but often beneﬁt signiﬁcantly from tuning of forty or more hyperparameters. Manual hyperparameter tuning can work very well when the user has a good starting point, such as one determined by others having worked on the same type of application and architecture, or when the user has months or years of experience in exploring hyperparameter values for neural networks applied to similar tasks. However, for many applications, these starting points are not available. In these cases, automated algorithms can ﬁnd useful values of the hyperparameters. If we think about the way in which the user of a learning algorithm searches for good values of the hyperparameters, we realize that an optimization is taking place: we are trying to ﬁnd a value of the hyperparameters that optimizes an objective function, such as validation error, sometimes under constraints (such as a budget for training time, memory or recognition time). It is therefore possible, in principle, to develop hyperparameter optimization algorithms that wrap a learning algorithm and choose its hyperparameters, thus hiding the hyperparameters of the learning algorithm from the user. Unfortunately, hyperparameter optimization algorithms often have their own hyperparameters, such as the range of values that should be explored for each of the learning algorithm’s hyperparameters. However, these secondary hyperparameters are usually easier to choose, in the sense that acceptable performance may be achieved on a wide range of tasks using the same secondary hyperparameters for all tasks.

11.4.3

Grid Search

When there are three or fewer hyperparameters, the common practice is to perform grid search. For each hyperparameter, the user selects a small ﬁnite set of values to explore. The grid search algorithm then trains a model for every joint speciﬁcation of hyperparameter values in the Cartesian product of the set of values for each individual hyperparameter. The experiment that yields the best validation set 434

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Grid

Random

Figure 11.2: Comparison of grid search and random search. For illustration purposes we display two hyperparameters but we are typically interested in having many more. (Left) To perform grid search, we provide a set of values for each hyperparameter. The search algorithm runs training for every joint hyperparameter setting in the cross product of these sets. (Right) To perform random search, we provide a probability distribution over joint hyperparameter conﬁgurations. Usually most of these hyperparameters are independent from each other. Common choices for the distribution over a single hyperparameter include uniform and log-uniform (to sample from a log-uniform distribution, take the exp of a sample from a uniform distribution). The search algorithm then randomly samples joint hyperparameter conﬁgurations and runs training with each of them. Both grid search and random search evaluate the validation set error and return the best conﬁguration. The ﬁgure illustrates the typical case where only some hyperparameters have a signiﬁcant inﬂuence on the result. In this illustration, only the hyperparameter on the horizontal axis has a signiﬁcant eﬀect. Grid search wastes an amount of computation that is exponential in the number of non-inﬂuential hyperparameters, while random search tests a unique value of every inﬂuential hyperparameter on nearly every trial.

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error is then chosen as having found the best hyperparameters. See the left of Fig. 11.2 for an illustration of a grid of hyperparameter values. How should the lists of values to search over be chosen? In the case of numerical (ordered) hyperparameters, the smallest and largest element of each list is chosen conservatively, based on prior experience with similar experiments, to make sure that the optimal value is very likely to be in the selected range. Typically, a grid search involves picking values approximately on a logarithmic scale, e.g., a learning rate taken within the set {. 1, .01, 10−3 , 10−4 , 10−5} , or a number of hidden units taken with the set {50, 100, 200, 500, 1000, 2000}.

Grid search usually performs best when it is performed repeatedly. For example, suppose that we ran a grid search over a hyperparameter α using values of {−1, 0, 1}. If the best value found is 1, then we underestimated the range in which the best α lies and we should shift the grid and run another search with α in, for example, { 1, 2, 3}. If we ﬁnd that the best value of α is 0, then we may wish to reﬁne our estimate by zooming in and running a grid search over {−.1, 0, .1}. The obvious problem with grid search is that its computational cost grows exponentially with the number of hyperparameters. If there are m hyperparameters, each taking at most n values, then the number of training and evaluation trials required grows as O (nm ). The trials may be run in parallel and exploit loose parallelism (with almost no need for communication between diﬀerent machines carrying out the search) Unfortunately, due to the exponential cost of grid search, even parallelization may not provide a satisfactory size of search.

11.4.4

Random Search

Fortunately, there is an alternative to grid search that is as simple to program, more convenient to use, and converges much faster to good values of the hyperparameters: random search (Bergstra and Bengio, 2012). A random search proceeds as follows. First we deﬁne a marginal distribution for each hyperparameter, e.g., a Bernoulli or multinoulli for binary or discrete hyperparameters, or a uniform distribution on a log-scale for positive real-valued hyperparameters. For example, log_learning_rate ∼ u(−1, −5)

learning_rate = 10log_learning_rate .

(11.2) (11.3)

where u(a, b ) indicates a sample of the uniform distribution in the interval (a, b). Similarly the log_number_of_hidden_units may be sampled from u(log(50), log(2000)). 436

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Unlike in the case of a grid search, one should not discretize or bin the values of the hyperparameters. This allows one to explore a larger set of values, and does not incur additional computational cost. In fact, as illustrated in Fig. 11.2, a random search can be exponentially more eﬃcient than a grid search, when there are several hyperparameters that do not strongly aﬀect the performance measure. This is studied at length in Bergstra and Bengio (2012), who found that random search reduces the validation set error much faster than grid search, in terms of the number of trials run by each method. As with grid search, one may often want to run repeated versions of random search, to reﬁne the search based on the results of the ﬁrst run. The main reason why random search ﬁnds good solutions faster than grid search is that the there are no wasted experimental runs, unlike in the case of grid search, when two values of a hyperparameter (given values of the other hyperparameters) would give the same result. In the case of grid search, the other hyperparameters would have the same values for these two runs, whereas with random search, they would usually have diﬀerent values. Hence if the change between these two values does not marginally make much diﬀerence in terms of validation set error, grid search will unnecessarily repeat two equivalent experiments while random search will still give two independent explorations of the other hyperparameters.

11.4.5

Model-Based Hyperparameter Optimization

The search for good hyperparameters can be cast as an optimization problem. The decision variables are the hyperparameters. The cost to be optimized is the validation set error that results from training using these hyperparameters. In simpliﬁed settings where it is feasible to compute the gradient of some diﬀerentiable error measure on the validation set with respect to the hyperparameters, we can simply follow this gradient (Bengio et al., 1999; Bengio, 2000; Maclaurin et al., 2015). Unfortunately, in most practical settings, this gradient is unavailable, either due to its high computation and memory cost, or due to hyperparameters having intrinsically non-diﬀerentiable interactions with the validation set error, as in the case of discrete-valued hyperparameters. To compensate for this lack of a gradient, we can build a model of the validation set error, then propose new hyperparameter guesses by performing optimization within this model. Most model-based algorithms for hyperparameter search use a Bayesian regression model to estimate both the expected value of the validation set error for each hyperparameter and the uncertainty around this expectation. Optimization thus involves a tradeoﬀ between exploration (proposing hyperparameters 437

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for which there is high uncertainty, which may lead to a large improvement but may also perform poorly) and exploitation (proposing hyperparameters which the model is conﬁdent will perform as well as any hyperparameters it has seen so far—usually hyperparameters that are very similar to ones it has seen before). Contemporary approaches to hyperparameter optimization include Spearmint (Snoek et al., 2012), TPE (Bergstra et al., 2011) and SMAC (Hutter et al., 2011). Currently, we cannot unambiguously recommend Bayesian hyperparameter optimization as an established tool for achieving better deep learning results or for obtaining those results with less eﬀort. Bayesian hyperparameter optimization sometimes performs comparably to human experts, sometimes better, but fails catastrophically on other problems. It may be worth trying to see if it works on a particular problem but is not yet suﬃciently mature or reliable. That being said, hyperparameter optimization is an important ﬁeld of research that, while often driven primarily by the needs of deep learning, holds the potential to beneﬁt not only the entire ﬁeld of machine learning but the discipline of engineering in general. One drawback common to most hyperparameter optimization algorithms with more sophistication than random search is that they require for a training experiment to run to completion before they are able to extract any information from the experiment. This is much less eﬃcient, in the sense of how much information can be gleaned early in an experiment, than manual search by a human practitioner, since one can usually tell early on if some set of hyperparameters is completely pathological. Swersky et al. (2014) have introduced an early version of an algorithm that maintains a set of multiple experiments. At various time points, the hyperparameter optimization algorithm can choose to begin a new experiment, to “freeze” a running experiment that is not promising, or to “thaw” and resume an experiment that was earlier frozen but now appears promising given more information.

11.5

Debugging Strategies

When a machine learning system performs poorly, it is usually diﬃcult to tell whether the poor performance is intrinsic to the algorithm itself or whether there is a bug in the implementation of the algorithm. Machine learning systems are diﬃcult to debug for a variety of reasons. In most cases, we do not know a priori what the intended behavior of the algorithm is. In fact, the entire point of using machine learning is that it will discover useful behavior that we were not able to specify ourselves. If we train a 438

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neural network on a new classiﬁcation task and it achieves 5% test error, we have no straightforward way of knowing if this is the expected behavior or sub-optimal behavior. A further diﬃculty is that most machine learning models have multiple parts that are each adaptive. If one part is broken, the other parts can adapt and still achieve roughly acceptable performance. For example, suppose that we are training a neural net with several layers parametrized by weights W and biases b. Suppose further that we have manually implemented the gradient descent rule for each parameter separately, and we made an error in the update for the biases: b← b−α

(11.4)

where α is the learning rate. This erroneous update does not use the gradient at all. It causes the biases to constantly become negative throughout learning, which is clearly not a correct implementation of any reasonable learning algorithm. The bug may not be apparent just from examining the output of the model though. Depending on the distribution of the input, the weights may be able to adapt to compensate for the negative biases. Most debugging strategies for neural nets are designed to get around one or both of these two diﬃculties. Either we design a case that is so simple that the correct behavior actually can be predicted, or we design a test that exercises one part of the neural net implementation in isolation. Some important debugging tests include: Visualize the model in action : When training a model to detect objects in images, view some images with the detections proposed by the model displayed superimposed on the image. When training a generative model of speech, listen to some of the speech samples it produces. This may seem obvious, but it is easy to fall into the practice of only looking at quantitative performance measurements like accuracy or log-likelihood. Directly observing the machine learning model performing its task will help you to determine whether the quantitative performance numbers it achieves seem reasonable. Evaluation bugs can be some of the most devastating bugs because they can mislead you into believing your system is performing well when it is not. Visualize the worst mistakes : Most models are able to output some sort of conﬁdence measure for the task they perform. For example, classiﬁers based on a softmax output layer assign a probability to each class. The probability assigned to the most likely class thus gives an estimate of the conﬁdence the model has in its classiﬁcation decision. Typically, maximum likelihood training results in these values being overestimates rather than accurate probabilities of correct prediction, 439

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but they are somewhat useful in the sense that examples that are actually less likely to be correctly labeled receive smaller probabilities under the model. By viewing the training set examples that are the hardest to model correctly, one can often discover problems with the way the data has been preprocessed or labeled. For example, the Street View transcription system originally had a problem where the address number detection system would crop the image too tightly and omit some of the digits. The transcription network then assigned very low probability to the correct answer on these images. Sorting the images to identify the most conﬁdent mistakes showed that there was a systematic problem with the cropping. Modifying the detection system to crop much wider images resulted in much better performance of the overall system, even though the transcription network needed to be able to process greater variation in the position and scale of the address numbers. Reasoning about software using train and test error: It is often diﬃcult to determine whether the underlying software is correctly implemented. Some clues can be obtained from the train and test error. If training error is low but test error is high, then it is likely that that the training procedure works correctly, and the model is overﬁtting for fundamental algorithmic reasons. An alternative possibility is that the test error is measured incorrectly due to a problem with saving the model after training then reloading it for test set evaluation, or if the test data was prepared diﬀerently from the training data. If both train and test error are high, then it is diﬃcult to determine whether there is a software defect or whether the model is underﬁtting due to fundamental algorithmic reasons. This scenario requires further tests, described next. Fit a tiny dataset: If you have high error on the training set, determine whether it is due to genuine underﬁtting or due to a software defect. Usually even small models can be guaranteed to be able ﬁt a suﬃciently small dataset. For example, a classiﬁcation dataset with only one example can be ﬁt just by setting the biases of the output layer correctly. Usually if you cannot train a classiﬁer to correctly label a single example, an autoencoder to successfully reproduce a single example with high ﬁdelity, or a generative model to consistently emit samples resembling a single example, there is a software defect preventing successful optimization on the training set. This test can be extended to a small dataset with few examples. Compare back-propagated derivatives to numerical derivatives: If you are using a software framework that requires you to implement your own gradient computations, or if you are adding a new operation to a diﬀerentiation library and must deﬁne its bprop method, then a common source of error is implementing this gradient expression incorrectly. One way to verify that these derivatives are correct 440

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is to compare the derivatives computed by your implementation of automatic diﬀerentiation to the derivatives computed by a ﬁnite diﬀerences. Because f (x) = lim

→0

f (x + ) − f (x) ,

(11.5)

we can approximate the derivative by using a small, ﬁnite : f (x) ≈

f (x + ) − f (x) .

(11.6)

We can improve the accuracy of the approximation by using the centered diﬀerence: f (x + 12 ) − f (x − 12 ) f (x ) ≈ .

(11.7)

The perturbation size must chosen to be large enough to ensure that the perturbation is not rounded down too much by ﬁnite-precision numerical computations. Usually, we will want to test the gradient or Jacobian of a vector-valued function g : R m → Rn . Unfortunately, ﬁnite diﬀerencing only allows us to take a single derivative at a time. We can either run ﬁnite diﬀerencing mn times to evaluate all of the partial derivatives of g, or we can apply the test to a new function that uses random projections at both the input and output of g. For example, we can apply our test of the implementation of the derivatives to f (x) where f (x) = uT g(vx), where u and v are randomly chosen vectors. Computing f (x) correctly requires being able to back-propagate through g correctly, yet is eﬃcient to do with ﬁnite diﬀerences because f has only a single input and a single output. It is usually a good idea to repeat this test for more than one value of u and v to reduce the chance that the test overlooks mistakes that are orthogonal to the random projection. If one has access to numerical computation on complex numbers, then there is a very eﬃcient way to numerically estimate the gradient by using complex numbers as input to the function (Squire and Trapp, 1998). The method is based on the observation that f (x + i) = f (x) + if (x) + O( 2) real(f (x + i)) = f (x) + O(2), imag(

f (x + i) ) = f (x) + O( 2 ),

(11.8) (11.9)

√ where i = −1. Unlike in the real-valued case above, there is no cancellation eﬀect due to taking the diﬀerence between the value of f at diﬀerent points. This allows the use of tiny values of like = 10 −150, which make the O(2) error insigniﬁcant for all practical purposes. 441

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Monitor histograms of activations and gradient: It is often useful to visualize statistics of neural network activations and gradients, collected over a large amount of training iterations (maybe one epoch). The pre-activation value of hidden units can tell us if the units saturate, or how often they do. For example, for rectiﬁers, how often are they oﬀ? Are there units that are always oﬀ? For tanh units, the average of the absolute value of the pre-activations tells us how saturated the unit is. In a deep network where the propagated gradients quickly grow or quickly vanish, optimization may be hampered. Finally, it is useful to compare the magnitude of parameter gradients to the magnitude of the parameters themselves. As suggested by Bottou (2015), we would like the magnitude of parameter updates over a minibatch to represent something like 1% of the magnitude of the parameter, not 50% or 0.001% (which would make the parameters move too slowly). It may be that some groups of parameters are moving at a good pace while others are stalled. When the data is sparse (like in natural language), some parameters may be very rarely updated, and this should be kept in mind when monitoring their evolution. Finally, many deep learning algorithms provide some sort of guarantee about the results produced at each step. For example, in Part III, we will see some approximate inference algorithms that work by using algebraic solutions to optimization problems. Typically these can be debugged by testing each of their guarantees. Some guarantees that some optimization algorithms oﬀer include that the objective function will never increase after one step of the algorithm, that the gradient with respect to some subset of variables will be zero after each step of the algorithm, and that the gradient with respect to all variables will be zero at convergence. Usually due to rounding error, these conditions will not hold exactly in a digital computer, so the debugging test should include some tolerance parameter.

11.6

Example: Multi-Digit Number Recognition

To provide an end-to-end description of how to apply our design methodology in practice, we present a brief account of the Street View transcription system, from the point of view of designing the deep learning components. Obviously, many other components of the complete system, such as the Street View cars, the database infrastructure, and so on, were of paramount importance. From the point of view of the machine learning task, the process began with data collection. The cars collected the raw data and human operators provided labels. The transcription task was preceded by a signiﬁcant amount of dataset 442

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curation, including using other machine learning techniques to detect the house numbers prior to transcribing them. The transcription project began with a choice of performance metrics and desired values for these metrics. An important general principle is to tailor the choice of metric to the business goals for the project. Because maps are only useful if they have high accuracy, it was important to set a high accuracy requirement for this project. Speciﬁcally, the goal was to obtain human-level, 98% accuracy. This level of accuracy may not always be feasible to obtain. In order to reach this level of accuracy, the Street View transcription system sacriﬁces coverage. Coverage thus became the main performance metric optimized during the project, with accuracy held at 98%. As the convolutional network improved, it became possible to reduce the conﬁdence threshold below which the network refuses to transcribe the input, eventually exceeding the goal of 95% coverage. After choosing quantitative goals, the next step in our recommended methodology is to rapidly establish a sensible baseline system. For vision tasks, this means a convolutional network with rectiﬁed linear units. The transcription project began with such a model. At the time, it was not common for a convolutional network to output a sequence of predictions. In order to begin with the simplest possible baseline, the ﬁrst implementation of the output layer of the model consisted of n diﬀerent softmax units to predict a sequence of n characters. These softmax units were trained exactly the same as if the task were classiﬁcation, with each softmax unit trained independently. Our recommended methodology is to iteratively reﬁne the baseline and test whether each change makes an improvement. The ﬁrst change to the Street View transcription system was motivated by a theoretical understanding of the coverage metric and the structure of the data. Speciﬁcally, the network refuses to classify an input x whenever the probability of the output sequence p(y | x ) < t for some threshold t. Initially, the deﬁnition of p(y | x) was ad-hoc, based on simply multiplying all of the softmax outputs together. This motivated the development of a specialized output layer and cost function that actually computed a principled log-likelihood. This approach allowed the example rejection mechanism to function much more eﬀectively. At this point, coverage was still below 90%, yet there were no obvious theoretical problems with the approach. Our methodology therefore suggests to instrument the train and test set performance in order to determine whether the problem is underﬁtting or overﬁtting. In this case, train and test set error were nearly identical. Indeed, the main reason this project proceeded so smoothly was the availability of a dataset with tens of millions of labeled examples. Because train 443

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and test set error were so similar, this suggested that the problem was either due to underﬁtting or due to a problem with the training data. One of the debugging strategies we recommend is to visualize the model’s worst errors. In this case, that meant visualizing the incorrect training set transcriptions that the model gave the highest conﬁdence. These proved to mostly consist of examples where the input image had been cropped too tightly, with some of the digits of the address being removed by the cropping operation. For example, a photo of an address “1849” might be cropped too tightly, with only the “849” remaining visible. This problem could have been resolved by spending weeks improving the accuracy of the address number detection system responsible for determining the cropping regions. Instead, the team took a much more practical decision, to simply expand the width of the crop region to be systematically wider than the address number detection system predicted. This single change added ten percentage points to the transcription system’s coverage. Finally, the last few percentage points of performance came from adjusting hyperparameters. This mostly consisted of making the model larger while maintaining some restrictions on its computational cost. Because train and test error remained roughly equal, it was always clear that any performance deﬁcits were due to underﬁtting, as well as due to a few remaining problems with the dataset itself. Overall, the transcription project was a great success, and allowed hundreds of millions of addresses to be transcribed both faster and at lower cost than would have been possible via human eﬀort. We hope that the design principles described in this chapter will lead to many other similar successes.

444

Chapter 12

Applications In this chapter, we describe how to use deep learning to solve applications in computer vision, speech recognition, natural language processing, and other application areas of commercial interest. We begin by discussing the large scale neural network implementations required for most serious AI applications. Next, we review several speciﬁc application areas that deep learning has been used to solve. While one goal of deep learning is to design algorithms that are capable of solving a broad variety of tasks, so far some degree of specialization is needed. For example, vision tasks require processing a large number of input features (pixels) per example. Language tasks require modeling a large number of possible values (words in the vocabulary) per input feature.

12.1

Large Scale Deep Learning

Deep learning is based on the philosophy of connectionism: while an individual biological neuron or an individual feature in a machine learning model is not intelligent, a large population of these neurons or features acting together can exhibit intelligent behavior. It truly is important to emphasize the fact that the number of neurons must be large. One of the key factors responsible for the improvement in neural network’s accuracy and the improvement of the complexity of tasks they can solve between the 1980s and today is the dramatic increase in the size of the networks we use. As we saw in Sec. 1.2.3, network sizes have grown exponentially for the past three decades, yet artiﬁcial neural networks are only as large as the nervous systems of insects. Because the size of neural networks is of paramount importance, deep learning 445

CHAPTER 12. APPLICATIONS

requires high performance hardware and software infrastructure.

12.1.1

Fast CPU Implementations

Traditionally, neural networks were trained using the CPU of a single machine. Today, this approach is generally considered insuﬃcient. We now mostly use GPU computing or the CPUs of many machines networked together. Before moving to these expensive setups, researchers worked hard to demonstrate that CPUs could not manage the high computational workload required by neural networks. A description of how to implement eﬃcient numerical CPU code is beyond the scope of this book, but we emphasize here that careful implementation for speciﬁc CPU families can yield large improvements. For example, in 2011, the best CPUs available could run neural network workloads faster when using ﬁxed-point arithmetic rather than ﬂoating-point arithmetic. By creating a carefully tuned ﬁxed-point implementation, Vanhoucke et al. (2011) obtained a 3× speedup over a strong ﬂoating-point system. Each new model of CPU has diﬀerent performance characteristics, so sometimes ﬂoating-point implementations can be faster too. The important principle is that careful specialization of numerical computation routines can yield a large payoﬀ. Other strategies, besides choosing whether to use ﬁxed or ﬂoating point, include optimizing data structures to avoid cache misses and using vector instructions. Many machine learning researchers neglect these implementation details, but when the performance of an implementation restricts the size of the model, the accuracy of the model suﬀers.

12.1.2

GPU Implementations

Most modern neural network implementations are based on graphics processing units. Graphics processing units (GPUs) are specialized hardware components that were originally developed for graphics applications. The consumer market for video gaming systems spurred development of graphics processing hardware. The performance characteristics needed for good video gaming systems turn out to be beneﬁcial for neural networks as well. Video game rendering requires performing many operations in parallel quickly. Models of characters and environments are speciﬁed in terms of lists of 3-D coordinates of vertices. Graphics cards must perform matrix multiplication and division on many vertices in parallel to convert these 3-D coordinates into 2-D on-screen coordinates. The graphics card must then perform many computations at each pixel in parallel to determine the color of each pixel. In both cases, the 446

CHAPTER 12. APPLICATIONS

computations are fairly simple and do not involve much branching compared to the computational workload that a CPU usually encounters. For example, each vertex in the same rigid object will be multiplied by the same matrix; there is no need to evaluate an if statement per-vertex to determine which matrix to multiply by. The computations are also entirely independent of each other, and thus may be parallelized easily. The computations also involve processing massive buﬀers of memory, containing bitmaps describing the texture (color pattern) of each object to be rendered. Together, this results in graphics cards having been designed to have a high degree of parallelism and high memory bandwidth, at the cost of having a lower clock speed and less branching capability relative to traditional CPUs. Neural network algorithms require the same performance characteristics as the real-time graphics algorithms described above. Neural networks usually involve large and numerous buﬀers of parameters, activation values, and gradient values, each of which must be completely updated during every step of training. These buﬀers are large enough to fall outside the cache of a traditional desktop computer so the memory bandwidth of the system often becomes the rate limiting factor. GPUs oﬀer a compelling advantage over CPUs due to their high memory bandwidth. Neural network training algorithms typically do not involve much branching or sophisticated control, so they are appropriate for GPU hardware. Since neural networks can be divided into multiple individual “neurons” that can be processed independently from the other neurons in the same layer, neural networks easily beneﬁt from the parallelism of GPU computing. GPU hardware was originally so specialized that it could only be used for graphics tasks. Over time, GPU hardware became more ﬂexible, allowing custom subroutines to be used to transform the coordinates of vertices or assign colors to pixels. In principle, there was no requirement that these pixel values actually be based on a rendering task. These GPUs could be used for scientiﬁc computing by writing the output of a computation to a buﬀer of pixel values. Steinkrau et al. (2005) implemented a two-layer fully connected neural network on a GPU and reported a 3X speedup over their CPU-based baseline. Shortly thereafter, Chellapilla et al. (2006) demonstrated that the same technique could be used to accelerate supervised convolutional networks. The popularity of graphics cards for neural network training exploded after the advent of general purpose GPUs. These GP-GPUs could execute arbitrary code, not just rendering subroutines. NVIDIA’s CUDA programming language provided a way to write this arbitrary code in a C-like language. With their relatively convenient programming model, massive parallelism, and high memory bandwidth, 447

CHAPTER 12. APPLICATIONS

GP-GPUs now oﬀer an ideal platform for neural network programming. This platform was rapidly adopted by deep learning researchers soon after it became available (Raina et al., 2009; Ciresan et al., 2010). Writing eﬃcient code for GP-GPUs remains a diﬃcult task best left to specialists. The techniques required to obtain good performance on GPU are very diﬀerent from those used on CPU. For example, good CPU-based code is usually designed to read information from the cache as much as possible. On GPU, most writable memory locations are not cached, so it can actually be faster to compute the same value twice, rather than compute it once and read it back from memory. GPU code is also inherently multi-threaded and the diﬀerent threads must be coordinated with each other carefully. For example, memory operations are faster if they can be coalesced. Coalesced reads or writes occur when several threads can each read or write a value that they need simultaneously, as part of a single memory transaction. Diﬀerent models of GPUs are able to coalesce diﬀerent kinds of read or write patterns. Typically, memory operations are easier to coalesce if among n threads, thread i accesses byte i + j of memory, and j is a multiple of some power of 2. The exact speciﬁcations diﬀer between models of GPU. Another common consideration for GPUs is making sure that each thread in a group executes the same instruction simultaneously. This means that branching can be diﬃcult on GPU. Threads are divided into small groups called warps. Each thread in a warp executes the same instruction during each cycle, so if diﬀerent threads within the same warp need to execute diﬀerent code paths, these diﬀerent code paths must be traversed sequentially rather than in parallel. Due to the diﬃculty of writing high performance GPU code, researchers should structure their workﬂow to avoid needing to write new GPU code in order to test new models or algorithms. Typically, one can do this by building a software library of high performance operations like convolution and matrix multiplication, then specifying models in terms of calls to this library of operations. For example, the machine learning library Pylearn2 (Goodfellow et al., 2013c) speciﬁes all of its machine learning algorithms in terms of calls to Theano (Bergstra et al., 2010; Bastien et al., 2012) and cuda-convnet (Krizhevsky, 2010), which provide these high-performance operations. This factored approach can also ease support for multiple kinds of hardware. For example, the same Theano program can run on either CPU or GPU, without needing to change any of the calls to Theano itself. Other libraries like TensorFlow (Abadi et al., 2015) and Torch (Collobert et al., 2011b) provide similar features.

448

CHAPTER 12. APPLICATIONS

12.1.3

Large Scale Distributed Implementations

In many cases, the computational resources available on a single machine are insuﬃcient. We therefore want to distribute the workload of training and inference across many machines. Distributing inference is simple, because each input example we want to process can be run by a separate machine. This is known as data parallelism. It is also possible to get model parallelism, where multiple machines work together on a single datapoint, with each machine running a diﬀerent part of the model. This is feasible for both inference and training. Data parallelism during training is somewhat harder. We can increase the size of the minibatch used for a single SGD step, but usually we get less than linear returns in terms of optimization performance. It would be better to allow multiple machines to compute multiple gradient descent steps in parallel. Unfortunately, the standard deﬁnition of gradient descent is as a completely sequential algorithm: the gradient at step t is a function of the parameters produced by step t − 1.

This can be solved using asynchronous stochastic gradient descent (Bengio et al., 2001; Recht et al., 2011). In this approach, several processor cores share the memory representing the parameters. Each core reads parameters without a lock, then computes a gradient, then increments the parameters without a lock. This reduces the average amount of improvement that each gradient descent step yields, because some of the cores overwrite each other’s progress, but the increased rate of production of steps causes the learning process to be faster overall. Dean et al. (2012) pioneered the multi-machine implementation of this lock-free approach to gradient descent, where the parameters are managed by a parameter server rather than stored in shared memory. Distributed asynchronous gradient descent remains the primary strategy for training large deep networks and is used by most major deep learning groups in industry (Chilimbi et al., 2014; Wu et al., 2015). Academic deep learning researchers typically cannot aﬀord the same scale of distributed learning systems but some research has focused on how to build distributed networks with relatively low-cost hardware available in the university setting (Coates et al., 2013).

12.1.4

Model Compression

In many commercial applications, it is much more important that the time and memory cost of running inference in a machine learning model be low than that the time and memory cost of training be low. For applications that do not require 449

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personalization, it is possible to train a model once, then deploy it to be used by billions of users. In many cases, the end user is more resource-constrained than the developer. For example, one might train a speech recognition network with a powerful computer cluster, then deploy it on mobile phones. A key strategy for reducing the cost of inference is model compression (Buciluˇa et al., 2006). The basic idea of model compression is to replace the original, expensive model with a smaller model that requires less memory and runtime to store and evaluate. Model compression is applicable when the size of the original model is driven primarily by a need to prevent overﬁtting. In most cases, the model with the lowest generalization error is an ensemble of several independently trained models. Evaluating all n ensemble members is expensive. Sometimes, even a single model generalizes better if it is large (for example, if it is regularized with dropout). These large models learn some function f(x), but do so using many more parameters than are necessary for the task. Their size is necessary only due to the limited number of training examples. As soon as we have ﬁt this function f(x), we can generate a training set containing inﬁnitely many examples, simply by applying f to randomly sampled points x. We then train the new, smaller, model to match f (x) on these points. In order to most eﬃciently use the capacity of the new, small model, it is best to sample the new x points from a distribution resembling the actual test inputs that will be supplied to the model later. This can be done by corrupting training examples or by drawing points from a generative model trained on the original training set. Alternatively, one can train the smaller model only on the original training points, but train it to copy other features of the model, such as its posterior distribution over the incorrect classes (Hinton et al., 2014, 2015).

12.1.5

Dynamic Structure

One strategy for accelerating data processing systems in general is to build systems that have dynamic structure in the graph describing the computation needed to process an input. Data processing systems can dynamically determine which subset of many neural networks should be run on a given input. Individual neural networks can also exhibit dynamic structure internally by determining which subset of features (hidden units) to compute given information from the input. This form of dynamic structure inside neural networks is sometimes called conditional computation (Bengio, 2013; Bengio et al., 2013b). Since many components of the architecture may be relevant only for a small amount of possible inputs, the system 450

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can run faster by computing these features only when they are needed. Dynamic structure of computations is a basic computer science principle applied generally throughout the software engineering discipline. The simplest versions of dynamic structure applied to neural networks are based on determining which subset of some group of neural networks (or other machine learning models) should be applied to a particular input. A venerable strategy for accelerating inference in a classiﬁer is to use a cascade of classiﬁers. The cascade strategy may be applied when the goal is to detect the presence of a rare object (or event). To know for sure that the object is present, we must use a sophisticated classiﬁer with high capacity, that is expensive to run. However, because the object is rare, we can usually use much less computation to reject inputs as not containing the object. In these situations, we can train a sequence of classiﬁers. The ﬁrst classiﬁers in the sequence have low capacity, and are trained to have high recall. In other words, they are trained to make sure we do not wrongly reject an input when the object is present. The ﬁnal classiﬁer is trained to have high precision. At test time, we run inference by running the classiﬁers in a sequence, abandoning any example as soon as any one element in the cascade rejects it. Overall, this allows us to verify the presence of objects with high conﬁdence, using a high capacity model, but does not force us to pay the cost of full inference for every example. There are two diﬀerent ways that the cascade can achieve high capacity. One way is to make the later members of the cascade individually have high capacity. In this case, the system as a whole obviously has high capacity, because some of its individual members do. It is also possible to make a cascade in which every individual model has low capacity but the system as a whole has high capacity due to the combination of many small models. Viola and Jones (2001) used a cascade of boosted decision trees to implement a fast and robust face detector suitable for use in handheld digital cameras. Their classiﬁer localizes a face using essentially a sliding window approach in which many windows are examined and rejected if they do not contain faces. Another version of cascades uses the earlier models to implement a sort of hard attention mechanism: the early members of the cascade localize an object and later members of the cascade perform further processing given the location of the object. For example, Google transcribes address numbers from Street View imagery using a two-step cascade that ﬁrst locates the address number with one machine learning model and then transcribes it with another (Goodfellow et al., 2014d). Decision trees themselves are an example of dynamic structure, because each node in the tree determines which of its subtrees should be evaluated for each input. A simple way to accomplish the union of deep learning and dynamic structure 451

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is to train a decision tree in which each node uses a neural network to make the splitting decision (Guo and Gelfand, 1992), though this has typically not been done with the primary goal of accelerating inference computations. In the same spirit, one can use a neural network, called the gater to select which one out of several expert networks will be used to compute the output, given the current input. The ﬁrst version of this idea is called the mixture of experts (Nowlan, 1990; Jacobs et al., 1991), in which the gater outputs a set of probabilities or weights (obtained via a softmax nonlinearity), one per expert, and the ﬁnal output is obtained by the weighted combination of the output of the experts. In that case, the use of the gater does not oﬀer a reduction in computational cost, but if a single expert is chosen by the gater for each example, we obtain the hard mixture of experts (Collobert et al., 2001, 2002), which can considerably accelerate training and inference time. This strategy works well when the number of gating decisions is small because it is not combinatorial. But when we want to select diﬀerent subsets of units or parameters, it is not possible to use a “soft switch” because it requires enumerating (and computing outputs for) all the gater conﬁgurations. To deal with this problem, several approaches have been explored to train combinatorial gaters. Bengio et al. (2013b) experiment with several estimators of the gradient on the gating probabilities, while Bacon et al. (2015) and Bengio et al. (2015a) use reinforcement learning techniques (policy gradient) to learn a form of conditional dropout on blocks of hidden units and get an actual reduction in computational cost without impacting negatively on the quality of the approximation. Another kind of dynamic structure is a switch, where a hidden unit can receive input from diﬀerent units depending on the context. This dynamic routing approach can be interpreted as an attention mechanism (Olshausen et al., 1993). So far, the use of a hard switch has not proven eﬀective on large-scale applications. Contemporary approaches instead use a weighted average over many possible inputs, and thus do not achieve all of the possible computational beneﬁts of dynamic structure. Contemporary attention mechanisms are described in Sec. 12.4.5.1. One major obstacle to using dynamically structured systems is the decreased degree of parallelism that results from the system following diﬀerent code branches for diﬀerent inputs. This means that few operations in the network can be described as matrix multiplication or batch convolution on a minibatch of examples. We can write more specialized sub-routines that convolve each example with diﬀerent kernels or multiply each row of a design matrix by a diﬀerent set of columns of weights. Unfortunately, these more specialized subroutines are diﬃcult to implement eﬃciently. CPU implementations will be slow due to the lack of cache coherence and GPU implementations will be slow due to the lack of coalesced 452

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memory transactions and the need to serialize warps when members of a warp take diﬀerent branches. In some cases, these issues can be mitigated by partitioning the examples into groups that all take the same branch, and processing these groups of examples simultaneously. This can be an acceptable strategy for minimizing the time required to process a ﬁxed amount of examples in an oﬄine setting. In a real-time setting where examples must be processed continuously, partitioning the workload can result in load-balancing issues. For example, if we assign one machine to process the ﬁrst step in a cascade and another machine to process the last step in a cascade, then the ﬁrst will tend to be overloaded and the last will tend to be underloaded. Similar issues arise if each machine is assigned to implement diﬀerent nodes of a neural decision tree.

12.1.6

Specialized Hardware Implementations of Deep Networks

Since the early days of neural networks research, hardware designers have worked on specialized hardware implementations that could speed up training and/or inference of neural network algorithms. See early and more recent reviews of specialized hardware for deep networks (Lindsey and Lindblad, 1994; Beiu et al., 2003; Misra and Saha, 2010). Diﬀerent forms of specialized hardware (Graf and Jackel, 1989; Mead and Ismail, 2012; Kim et al., 2009; Pham et al., 2012; Chen et al., 2014a,b) have been developed over the last decades, either with ASICs (application-speciﬁc integrated circuit), either with digital (based on binary representations of numbers), analog (Graf and Jackel, 1989; Mead and Ismail, 2012) (based on physical implementations of continuous values as voltages or currents) or hybrid implementations (combining digital and analog components). In recent years more ﬂexible FPGA (ﬁeld programmable gated array) implementations (where the particulars of the circuit can be written on the chip after it has been built) have been developed. Though software implementations on general-purpose processing units (CPUs and GPUs) typically use 32 or 64 bits of precision to represent ﬂoating point numbers, it has long been known that it was possible to use less precision, at least at inference time (Holt and Baker, 1991; Holi and Hwang, 1993; Presley and Haggard, 1994; Simard and Graf, 1994; Wawrzynek et al., 1996; Savich et al., 2007). This has become a more pressing issue in recent years as deep learning has gained in popularity in industrial products, and as the great impact of faster hardware was demonstrated with GPUs. Another factor that motivates current research on specialized hardware for deep networks is that the rate of progress of a single CPU or GPU core has slowed down, and most recent improvements in computing speed have come from parallelization across cores (either in CPUs or 453

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GPUs). This is very diﬀerent from the situation of the 1990s (the previous neural network era) where the hardware implementations of neural networks (which might take two years from inception to availability of a chip) could not keep up with the rapid progress and low prices of general-purpose CPUs. Building specialized hardware is thus a way to push the envelope further, at a time when new hardware designs are being developed for low-power devices such as phones, aiming for general-public applications of deep learning (e.g., with speech, computer vision or natural language). Recent work on low-precision implementations of backprop-based neural nets (Vanhoucke et al., 2011; Courbariaux et al., 2015; Gupta et al., 2015) suggests that between 8 and 16 bits of precision can suﬃce for using or training deep neural networks with back-propagation. What is clear is that more precision is required during training than at inference time, and that some forms of dynamic ﬁxed point representation of numbers can be used to reduce how many bits are required per number. Traditional ﬁxed point numbers are restricted to a ﬁxed range (which corresponds to a given exponent in a ﬂoating point representation). Dynamic ﬁxed point representations share that range among a set of numbers (such as all the weights in one layer). Using ﬁxed point rather than ﬂoating point representations and using less bits per number reduces the hardware surface area, power requirements and computing time needed for performing multiplications, and multiplications are the most demanding of the operations needed to use or train a modern deep network with backprop.

12.2

Computer Vision

Computer vision has traditionally been one of the most active research areas for deep learning applications, because vision is a task that is eﬀortless for humans and many animals but challenging for computers (Ballard et al., 1983). Many of the most popular standard benchmark tasks for deep learning algorithms are forms of object recognition or optical character recognition. Computer vision is a very broad ﬁeld encompassing a wide variety of ways of processing images, and an amazing diversity of applications. Applications of computer vision range from reproducing human visual abilities, such as recognizing faces, to creating entirely new categories of visual abilities. As an example of the latter category, one recent computer vision application is to recognize sound waves from the vibrations they induce in objects visible in a video (Davis et al., 2014). Most deep learning research on computer vision has not focused on such exotic applications that expand the realm of what is possible with imagery but 454

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rather a small core of AI goals aimed at replicating human abilities. Most deep learning for computer vision is used for object recognition or detection of some form, whether this means reporting which object is present in an image, annotating an image with bounding boxes around each object, transcribing a sequence of symbols from an image, or labeling each pixel in an image with the identity of the object it belongs to. Because generative modeling has been a guiding principle of deep learning research, there is also a large body of work on image synthesis using deep models. While image synthesis ex nihilo is usually not considered a computer vision endeavor, models capable of image synthesis are usually useful for image restoration, a computer vision task involving repairing defects in images or removing objects from images.

12.2.1

Preprocessing

Many application areas require sophisticated preprocessing because the original input comes in a form that is diﬃcult for many deep learning architectures to represent. Computer vision usually requires relatively little of this kind of preprocessing. The images should be standardized so that their pixels all lie in the same, reasonable range, like [0,1] or [-1, 1]. Mixing images that lie in [0,1] with images that lie in [0, 255] will usually result in failure. Formatting images to have the same scale is the only kind of preprocessing that is strictly necessary. Many computer vision architectures require images of a standard size, so images must be cropped or scaled to ﬁt that size. However, even this rescaling is not always strictly necessary. Some convolutional models accept variably-sized inputs and dynamically adjust the size of their pooling regions to keep the output size constant (Waibel et al., 1989). Other convolutional models have variable-sized output that automatically scales in size with the input, such as models that denoise or label each pixel in an image (Hadsell et al., 2007). Dataset augmentation may be seen as a way of preprocessing the training set only. Dataset augmentation is an excellent way to reduce the generalization error of most computer vision models. A related idea applicable at test time is to show the model many diﬀerent versions of the same input (for example, the same image cropped at slightly diﬀerent locations) and have the diﬀerent instantiations of the model vote to determine the output. This latter idea can be interpreted as an ensemble approach, and helps to reduce generalization error. Other kinds of preprocessing are applied to both the train and the test set with the goal of putting each example into a more canonical form in order to reduce the amount of variation that the model needs to account for. Reducing the amount of variation in the data can both reduce generalization error and reduce the size of 455

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the model needed to ﬁt the training set. Simpler tasks may be solved by smaller models, and simpler solutions are more likely to generalize well. Preprocessing of this kind is usually designed to remove some kind of variability in the input data that is easy for a human designer to describe and that the human designer is conﬁdent has no relevance to the task. When training with large datasets and large models, this kind of preprocessing is often unnecessary, and it is best to just let the model learn which kinds of variability it should become invariant to. For example, the AlexNet system for classifying ImageNet only has one preprocessing step: subtracting the mean across training examples of each pixel (Krizhevsky et al., 2012). 12.2.1.1

Contrast Normalization

One of the most obvious sources of variation that can be safely removed for many tasks is the amount of contrast in the image. Contrast simply refers to the magnitude of the diﬀerence between the bright and the dark pixels in an image. There are many ways of quantifying the contrast of an image. In the context of deep learning, contrast usually refers to the standard deviation of the pixels in an image or region of an image. Suppose we have an image represented by a tensor X ∈ R r×c×3 , with X i,j,1 being the red intensity at row i and column j, Xi,j,2 giving the green intensity and Xi,j,3 giving the blue intensity. Then the contrast of the entire image is given by r c 3 1 ¯ 2 (12.1) X i,j,k − X 3rc i=1 j=1 k=1

¯ is the mean intensity of the entire image: where X r

c

3

¯= 1 Xi,j,k . X 3rc i=1 j=1 k=1

(12.2)

Global contrast normalization (GCN) aims to prevent images from having varying amounts of contrast by subtracting the mean from each image, then rescaling it so that the standard deviation across its pixels is equal to some constant s. This approach is complicated by the fact that no scaling factor can change the contrast of a zero-contrast image (one whose pixels all have equal intensity). Images with very low but non-zero contrast often have little information content. Dividing by the true standard deviation usually accomplishes nothing more than amplifying sensor noise or compression artifacts in such cases. This 456

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motivates introducing a small, positive regularization parameter λ to bias the estimate of the standard deviation. Alternately, one can constrain the denominator to be at least . Given an input image X, GCN produces an output image X , deﬁned such that ¯ Xi,j,k − X . =s (12.3) Xi,j,k 2 c 3 1 r ¯ max , λ + 3rc i=1 j=1 k=1 Xi,j,k − X

Datasets consisting of large images cropped to interesting objects are unlikely to contain any images with nearly constant intensity. In these cases, it is safe to practically ignore the small denominator problem by setting λ = 0 and avoid division by 0 in extremely rare cases by setting to an extremely low value like 10 −8 . This is the approach used by Goodfellow et al. (2013a) on the CIFAR-10 dataset. Small images cropped randomly are more likely to have nearly constant intensity, making aggressive regularization more useful. Coates et al. (2011) used = 0 and λ = 10 on small, randomly selected patches drawn from CIFAR-10.

The scale parameter s can usually be set to 1, as done by Coates et al. (2011), or chosen to make each individual pixel have standard deviation across examples close to 1, as done by Goodfellow et al. (2013a). The standard deviation in Eq. 12.3 is just a rescaling of the L2 norm of the image (assuming the mean of the image has already been removed). It is preferable to deﬁne GCN in terms of standard deviation rather than L 2 norm because the standard deviation includes division by the number of pixels, so GCN based on standard deviation allows the same s to be used regardless of image size. However, the observation that the L 2 norm is proportional to the standard deviation can help build a useful intuition. One can understand GCN as mapping examples to a spherical shell. See Fig. 12.1 for an illustration. This can be a useful property because neural networks are often better at responding to directions in space rather than exact locations. Responding to multiple distances in the same direction requires hidden units with collinear weight vectors but diﬀerent biases. Such coordination can be diﬃcult for the learning algorithm to discover. Additionally, many shallow graphical models have problems with representing multiple separated modes along the same line. GCN avoids these problems by reducing each example to a direction rather than a direction and a distance. Counterintuitively, there is a preprocessing operation known as sphering and it is not the same operation as GCN. Sphering does not refer to making the data lie on a spherical shell, but rather to rescaling the principal components to have equal variance, so that the multivariate normal distribution used by PCA has spherical contours. Sphering is more commonly known as whitening. 457

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Raw input

GCN, λ = 0

GCN, λ = 10−2

x1

1.5

0.0

−1.5 −1.5

0.0 x0

1.5

−1.5

0.0 x0

1.5

−1.5

0.0 x0

1.5

Figure 12.1: GCN maps examples onto a sphere. (Left) Raw input data may have any norm. (Center) GCN with λ = 0 maps all non-zero examples perfectly onto a sphere. Here we use s = 1 and = 10−8 . Because we use GCN based on normalizing the standard deviation rather than the L2 norm, the resulting sphere is not the unit sphere. (Right) Regularized GCN, with λ > 0, draws examples toward the sphere but does not completely discard the variation in their norm. We leave s and the same as before.

Global contrast normalization will often fail to highlight image features we would like to stand out, such as edges and corners. If we have a scene with a large dark area and a large bright area (such as a city square with half the image in the shadow of a building) then global contrast normalization will ensure there is a large diﬀerence between the brightness of the dark area and the brightness of the light area. It will not, however, ensure that edges within the dark region stand out. This motivates local contrast normalization. Local contrast normalization ensures that the contrast is normalized across each small window, rather than over the image as a whole. See Fig. 12.2 for a comparison of global and local contrast normalization. Various deﬁnitions of local contrast normalization are possible. In all cases, one modiﬁes each pixel by subtracting a mean of nearby pixels and dividing by a standard deviation of nearby pixels. In some cases, this is literally the mean and standard deviation of all pixels in a rectangular window centered on the pixel to be modiﬁed (Pinto et al., 2008). In other cases, this is a weighted mean and weighted standard deviation using Gaussian weights centered on the pixel to be modiﬁed. In the case of color images, some strategies process diﬀerent color channels separately while others combine information from diﬀerent channels to normalize each pixel (Sermanet et al., 2012). 458

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Input image

GCN

LCN

Figure 12.2: A comparison of global and local contrast normalization. Visually, the eﬀects of global contrast normalization are subtle. It places all images on roughly the same scale, which reduces the burden on the learning algorithm to handle multiple scales. Local contrast normalization modiﬁes the image much more, discarding all regions of constant intensity. This allows the model to focus on just the edges. Regions of ﬁne texture, such as the houses in the second row, may lose some detail due to the bandwidth of the normalization kernel being too high.

Local contrast normalization can usually be implemented eﬃciently by using separable convolution (see Sec. 9.8) to compute feature maps of local means and local standard deviations, then using element-wise subtraction and element-wise division on diﬀerent feature maps. Local contrast normalization is a diﬀerentiable operation and can also be used as a nonlinearity applied to the hidden layers of a network, as well as a preprocessing operation applied to the input. As with global contrast normalization, we typically need to regularize local contrast normalization to avoid division by zero. In fact, because local contrast normalization typically acts on smaller windows, it is even more important to regularize. Smaller windows are more likely to contain values that are all nearly the same as each other, and thus more likely to have zero standard deviation. 12.2.1.2

Dataset Augmentation

As described in Sec. 7.4, it is easy to improve the generalization of a classiﬁer by increasing the size of the training set by adding extra copies of the training examples that have been modiﬁed with transformations that do not change the 459

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class. Object recognition is a classiﬁcation task that is especially amenable to this form of dataset augmentation because the class is invariant to so many transformations and the input can be easily transformed with many geometric operations. As described before, classiﬁers can beneﬁt from random translations, rotations, and in some cases, ﬂips of the input to augment the dataset. In specialized computer vision applications, more advanced transformations are commonly used for dataset augmentation. These schemes include random perturbation of the colors in an image (Krizhevsky et al., 2012) and nonlinear geometric distortions of the input (LeCun et al., 1998b).

12.3

Speech Recognition

The task of speech recognition is to map an acoustic signal containing a spoken natural language utterance into the corresponding sequence of words intended by the speaker. Let X = (x(1) , x(2) , . . . , x(T )) denote the sequence of acoustic input vectors (traditionally produced by splitting the audio into 20ms frames). Most speech recognition systems preprocess the input using specialized hand-designed features, but some (Jaitly and Hinton, 2011) deep learning systems learn features from raw input. Let y = (y1 , y2 , . . . , yN ) denote the target output sequence (usually a sequence of words or characters). The automatic speech recognition (ASR) task ∗ consists of creating a function fASR that computes the most probable linguistic sequence y given the acoustic sequence X : f ∗ASR(X ) = arg max P ∗ (y | X = X )

(12.4)

y

where P ∗ is the true conditional distribution relating the inputs X to the targets y. Since the 1980s and until about 2009–2012, state-of-the art speech recognition systems primarily combined hidden Markov models (HMMs) and Gaussian mixture models (GMMs). GMMs modeled the association between acoustic features and phonemes (Bahl et al., 1987), while HMMs modeled the sequence of phonemes. The GMM-HMM model family treats acoustic waveforms as being generated by the following process: ﬁrst an HMM generates a sequence of phonemes and discrete sub-phonemic states (such as the beginning, middle, and end of each phoneme), then a GMM transforms each discrete symbol into a brief segment of audio waveform. Although GMM-HMM systems dominated ASR until recently, speech recognition was actually one of the ﬁrst areas where neural networks were applied, and numerous ASR systems from the late 1980s and early 1990s used 460

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neural nets (Bourlard and Wellekens, 1989; Waibel et al., 1989; Robinson and Fallside, 1991; Bengio et al., 1991, 1992; Konig et al., 1996). At the time, the performance of ASR based on neural nets approximately matched the performance of GMM-HMM systems. For example, Robinson and Fallside (1991) achieved 26% phoneme error rate on the TIMIT (Garofolo et al., 1993) corpus (with 39 phonemes to discriminate between), which was better than or comparable to HMM-based systems. Since then, TIMIT has been a benchmark for phoneme recognition, playing a role similar to the role MNIST plays for object recognition. However, because of the complex engineering involved in software systems for speech recognition and the eﬀort that had been invested in building these systems on the basis of GMM-HMMs, the industry did not see a compelling argument for switching to neural networks. As a consequence, until the late 2000s, both academic and industrial research in using neural nets for speech recognition mostly focused on using neural nets to learn extra features for GMM-HMM systems. Later, with much larger and deeper models and much larger datasets, recognition accuracy was dramatically improved by using neural networks to replace GMMs for the task of associating acoustic features to phonemes (or subphonemic states). Starting in 2009, speech researchers applied a form of deep learning based on unsupervised learning to speech recognition. This approach to deep learning was based on training undirected probabilistic models called restricted Boltzmann machines (RBMs) to model the input data. RBMs will be described in Part III. To solve speech recognition tasks, unsupervised pretraining was used to build deep feedforward networks whose layers were each initialized by training an RBM. These networks take spectral acoustic representations in a ﬁxed-size input window (around a center frame) and predict the conditional probabilities of HMM states for that center frame. Training such deep networks helped to signiﬁcantly improve the recognition rate on TIMIT (Mohamed et al., 2009, 2012a), bringing down the phoneme error rate from about 26% to 20.7%. See Mohamed et al. (2012b) for an analysis of reasons for the success of these models. Extensions to the basic phone recognition pipeline included the addition of speaker-adaptive features (Mohamed et al., 2011) that further reduced the error rate. This was quickly followed up by work to expand the architecture from phoneme recognition (which is what TIMIT is focused on) to large-vocabulary speech recognition (Dahl et al., 2012), which involves not just recognizing phonemes but also recognizing sequences of words from a large vocabulary. Deep networks for speech recognition eventually shifted from being based on pretraining and Boltzmann machines to being based on techniques such as rectiﬁed linear units and dropout (Zeiler et al., 2013; Dahl et al., 2013). By that time, several of the major speech groups in industry had started exploring deep learning in collaboration with 461

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academic researchers. Hinton et al. (2012a) describe the breakthroughs achieved by these collaborators, which are now deployed in products such as mobile phones. Later, as these groups explored larger and larger labeled datasets and incorporated some of the methods for initializing, training, and setting up the architecture of deep nets, they realized that the unsupervised pretraining phase was either unnecessary or did not bring any signiﬁcant improvement. These breakthroughs in recognition performance for word error rate in speech recognition were unprecedented (around 30% improvement) and were following a long period of about ten years during which error rates did not improve much with the traditional GMM-HMM technology, in spite of the continuously growing size of training sets (see Fig. 2.4 of Deng and Yu (2014)). This created a rapid shift in the speech recognition community towards deep learning. In a matter of roughly two years, most of the industrial products for speech recognition incorporated deep neural networks and this success spurred a new wave of research into deep learning algorithms and architectures for ASR, which is still ongoing today. One of these innovations was the use of convolutional networks (Sainath et al., 2013) that replicate weights across time and frequency, improving over the earlier time-delay neural networks that replicated weights only across time. The new two-dimensional convolutional models regard the input spectrogram not as one long vector but as an image, with one axis corresponding to time and the other to frequency of spectral components. Another important push, still ongoing, has been towards end-to-end deep learning speech recognition systems that completely remove the HMM. The ﬁrst major breakthrough in this direction came from Graves et al. (2013) who trained a deep LSTM RNN (see Sec. 10.10), using MAP inference over the frame-to-phoneme alignment, as in LeCun et al. (1998b) and in the CTC framework (Graves et al., 2006; Graves, 2012). A deep RNN (Graves et al., 2013) has state variables from several layers at each time step, giving the unfolded graph two kinds of depth: ordinary depth due to a stack of layers, and depth due to time unfolding. This work brought the phoneme error rate on TIMIT to a record low of 17.7%. See Pascanu et al. (2014a) and Chung et al. (2014) for other variants of deep RNNs, applied in other settings. Another contemporary step toward end-to-end deep learning ASR is to let the system learn how to “align” the acoustic-level information with the phonetic-level information (Chorowski et al., 2014; Lu et al., 2015).

462

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12.4

Natural Language Processing

Natural language processing (NLP) is the use of human languages, such as English or French, by a computer. Computer programs typically read and emit specialized languages designed to allow eﬃcient and unambiguous parsing by simple programs. More naturally occurring languages are often ambiguous and defy formal description. Natural language processing includes applications such as machine translation, in which the learner must read a sentence in one human language and emit an equivalent sentence in another human language. Many NLP applications are based on language models that deﬁne a probability distribution over sequences of words, characters or bytes in a natural language. As with the other applications discussed in this chapter, very generic neural network techniques can be successfully applied to natural language processing. However, to achieve excellent performance and to scale well to large applications, some domain-speciﬁc strategies become important. To build an eﬃcient model of natural language, we must usually use techniques that are specialized for processing sequential data. In many cases, we choose to regard natural language as a sequence of words, rather than a sequence of individual characters or bytes. Because the total number of possible words is so large, word-based language models must operate on an extremely high-dimensional and sparse discrete space. Several strategies have been developed to make models of such a space eﬃcient, both in a computational and in a statistical sense.

12.4.1

n-grams

A language model deﬁnes a probability distribution over sequences of tokens in a natural language. Depending on how the model is designed, a token may be a word, a character, or even a byte. Tokens are always discrete entities. The earliest successful language models were based on models of ﬁxed-length sequences of tokens called n-grams. An n-gram is a sequence of n tokens. Models based on n-grams deﬁne the conditional probability of the n-th token given the preceding n − 1 tokens. The model uses products of these conditional distributions to deﬁne the probability distribution over longer sequences: τ P (x 1, . . . , xτ ) = P (x1 , . . . , xn−1) P (x t | x t−n+1 , . . . , xt−1 ). (12.5) t=n

This decomposition is justiﬁed by the chain rule of probability. The probability distribution over the initial sequence P (x1, . . . , xn−1) may be modeled by a diﬀerent model with a smaller value of n. 463

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Training n-gram models is straightforward because the maximum likelihood estimate can be computed simply by counting how many times each possible n gram occurs in the training set. Models based on n-grams have been the core building block of statistical language modeling for many decades (Jelinek and Mercer, 1980; Katz, 1987; Chen and Goodman, 1999). For small values of n , models have particular names: unigram for n=1, bigram for n=2, and trigram for n=3. These names derive from the Latin preﬁxes for the corresponding numbers and the Greek suﬃx “-gram” denoting something that is written. Usually we train both an n-gram model and an n−1 gram model simultaneously. This makes it easy to compute P (x t | x t−n+1 , . . . , xt ) =

P n (xt−n+1 , . . . , xt ) P n−1(xt−n+1 , . . . , xt−1)

(12.6)

simply by looking up two stored probabilities. For this to exactly reproduce inference in P n, we must omit the ﬁnal character from each sequence when we train P n−1. As an example, we demonstrate how a trigram model computes the probability of the sentence “THE DOG RAN AWAY.” The ﬁrst words of the sentence cannot be handled by the default formula based on conditional probability because there is no context at the beginning of the sentence. Instead, we must use the marginal probability over words at the start of the sentence. We thus evaluate P3 (THE DOG RAN). Finally, the last word may be predicted using the typical case, of using the conditional distribution P (AWAY | DOG RAN). Putting this together with Eq. 12.6, we obtain: P (THE DOG RAN AWAY) = P3 (THE DOG RAN)P3 (DOG RAN AWAY)/P2 (DOG RAN). (12.7) A fundamental limitation of maximum likelihood for n-gram models is that Pn as estimated from training set counts is very likely to be zero in many cases, even though the tuple (x t−n+1, . . . , xt ) may appear in the test set. This can cause two diﬀerent kinds of catastrophic outcomes. When P n−1 is zero, the ratio is undeﬁned, so the model does not even produce a sensible output. When Pn−1 is non-zero but Pn is zero, the test log-likelihood is −∞. To avoid such catastrophic outcomes, most n-gram models employ some form of smoothing. Smoothing techniques shift probability mass from the observed tuples to unobserved ones that are similar. See Chen and Goodman (1999) for a review and empirical comparisons. One basic technique consists of adding non-zero probability mass to all of the possible next symbol values. This method can be justiﬁed as Bayesian inference with a uniform 464

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or Dirichlet prior over the count parameters. Another very popular idea is to form a mixture model containing higher-order and lower-order n-gram models, with the higher-order models providing more capacity and the lower-order models being more likely to avoid counts of zero. Back-oﬀ methods look-up the lower-order n-grams if the frequency of the context xt−1, . . . , x t−n+1 is too small to use the higher-order model. More formally, they estimate the distribution over xt by using contexts x t−n+k, . . . , xt−1 , for increasing k, until a suﬃciently reliable estimate is found. Classical n-gram models are particularly vulnerable to the curse of dimensionality. There are |V|n possible n-grams and |V| is often very large. Even with a massive training set and modest n, most n-grams will not occur in the training set. One way to view a classical n-gram model is that it is performing nearest-neighbor lookup. In other words, it can be viewed as a local non-parametric predictor, similar to k-nearest neighbors. The statistical problems facing these extremely local predictors are described in Sec. 5.11.2. The problem for a language model is even more severe than usual, because any two diﬀerent words have the same distance from each other in one-hot vector space. It is thus diﬃcult to leverage much information from any “neighbors”—only training examples that repeat literally the same context are useful for local generalization. To overcome these problems, a language model must be able to share knowledge between one word and other semantically similar words. To improve the statistical eﬃciency of n-gram models, class-based language models (Brown et al., 1992; Ney and Kneser, 1993; Niesler et al., 1998) introduce the notion of word categories and then share statistical strength between words that are in the same category. The idea is to use a clustering algorithm to partition the set of words into clusters or classes, based on their co-occurrence frequencies with other words. The model can then use word class IDs rather than individual word IDs to represent the context on the right side of the conditioning bar. Composite models combining word-based and class-based models via mixing or back-oﬀ are also possible. Although word classes provide a way to generalize between sequences in which some word is replaced by another of the same class, much information is lost in this representation.

12.4.2

Neural Language Models

Neural language models or NLMs are a class of language model designed to overcome the curse of dimensionality problem for modeling natural language sequences by using a distributed representation of words (Bengio et al., 2001). Unlike classbased n-gram models, neural language models are able to recognize that two words 465

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are similar without losing the ability to encode each word as distinct from the other. Neural language models share statistical strength between one word (and its context) and other similar words and contexts. The distributed representation the model learns for each word enables this sharing by allowing the model to treat words that have features in common similarly. For example, if the word dog and the word cat map to representations that share many attributes, then sentences that contain the word cat can inform the predictions that will be made by the model for sentences that contain the word dog, and vice-versa. Because there are many such attributes, there are many ways in which generalization can happen, transferring information from each training sentence to an exponentially large number of semantically related sentences. The curse of dimensionality requires the model to generalize to a number of sentences that is exponential in the sentence length. The model counters this curse by relating each training sentence to an exponential number of similar sentences. We sometimes call these word representations word embeddings. In this interpretation, we view the raw symbols as points in a space of dimension equal to the vocabulary size. The word representations embed those points in a feature space of lower dimension. In the original space, every word is √ represented by a one-hot vector, so every pair of words is at Euclidean distance 2 from each other. In the embedding space, words that frequently appear in similar contexts (or any pair of words sharing some “features” learned by the model) are close to each other. This often results in words with similar meanings being neighbors. Fig. 12.3 zooms in on speciﬁc areas of a learned word embedding space to show how semantically similar words map to representations that are close to each other. Neural networks in other domains also deﬁne embeddings. For example, a hidden layer of a convolutional network provides an “image embedding.” Usually NLP practitioners are much more interested in this idea of embeddings because natural language does not originally lie in a real-valued vector space. The hidden layer has provided a more qualitatively dramatic change in the way the data is represented. The basic idea of using distributed representations to improve models for natural language processing is not restricted to neural networks. It may also be used with graphical models that have distributed representations in the form of multiple latent variables (Mnih and Hinton, 2007).

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−6 −7 −8

22 France China Russian French English

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17 35.0 35.5 36.0 36.5 37.0 37.5 38.0

Figure 12.3: Two-dimensional visualizations of word embeddings obtained from a neural machine translation model (Bahdanau et al., 2015), zooming in on speciﬁc areas where semantically related words have embedding vectors that are close to each other. Countries appear on the left and numbers on the right. Keep in mind that these embeddings are 2-D for the purpose of visualization. In real applications, embeddings typically have higher dimensionality and can simultaneously capture many kinds of similarity between words.

12.4.3

High-Dimensional Outputs

In many natural language applications, we often want our models to produce words (rather than characters) as the fundamental unit of the output. For large vocabularies, it can be very computationally expensive to represent an output distribution over the choice of a word, because the vocabulary size is large. In many applications, V contains hundreds of thousands of words. The naive approach to representing such a distribution is to apply an aﬃne transformation from a hidden representation to the output space, then apply the softmax function. Suppose we have a vocabulary V with size |V|. The weight matrix describing the linear component of this aﬃne transformation is very large, because its output dimension is |V|. This imposes a high memory cost to represent the matrix, and a high computational cost to multiply by it. Because the softmax is normalized across all |V| outputs, it is necessary to perform the full matrix multiplication at training time as well as test time—we cannot calculate only the dot product with the weight vector for the correct output. The high computational costs of the output layer thus arise both at training time (to compute the likelihood and its gradient) and at test time (to compute probabilities for all or selected words). For specialized loss functions, the gradient can be computed eﬃciently (Vincent et al., 2015), but the standard cross-entropy loss applied to a traditional softmax output layer poses 467

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many diﬃculties. Suppose that h is the top hidden layer used to predict the output probabilities y. ˆ If we parametrize the transformation from h to yˆ with learned weights W and learned biases b, then the aﬃne-softmax output layer performs the following computations: ai = bi + W ij hj ∀i ∈ {1, . . . , |V|}, (12.8) j a ei

yˆi = |V|

i =1

ea i

(12.9)

.

If h contains nh elements then the above operation is O(|V|nh). With nh in the thousands and |V| in the hundreds of thousands, this operation dominates the computation of most neural language models. 12.4.3.1

Use of a Short List

The ﬁrst neural language models (Bengio et al., 2001, 2003) dealt with the high cost of using a softmax over a large number of output words by limiting the vocabulary size to 10,000 or 20,000 words. Schwenk and Gauvain (2002) and Schwenk (2007) built upon this approach by splitting the vocabulary V into a shortlist L of most frequent words (handled by the neural net) and a tail T = V\L of more rare words (handled by an n-gram model). To be able to combine the two predictions, the neural net also has to predict the probability that a word appearing after context C belongs to the tail list. This may be achieved by adding an extra sigmoid output unit to provide an estimate of P (i ∈ T | C ). The extra output can then be used to achieve an estimate of the probability distribution over all words in V as follows: P (y = i | C ) =1i∈L P (y = i | C, i ∈ L)(1 − P (i ∈ T | C )) + 1 i∈T P (y = i | C, i ∈ T)P (i ∈ T | C )

(12.10) (12.11)

where P (y = i | C, i ∈ L) is provided by the neural language model and P (y = i | C, i ∈ T) is provided by the n-gram model. With slight modiﬁcation, this approach can also work using an extra output value in the neural language model’s softmax layer, rather than a separate sigmoid unit. An obvious disadvantage of the short list approach is that the potential generalization advantage of the neural language models is limited to the most frequent words, where, arguably, it is the least useful. This disadvantage has stimulated the exploration of alternative methods to deal with high-dimensional outputs, described below. 468

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12.4.3.2

Hierarchical Softmax

A classical approach (Goodman, 2001) to reducing the computational burden of high-dimensional output layers over large vocabulary sets V is to decompose probabilities hierarchically. Instead of necessitating a number of computations proportional to |V| (and also proportional to the number of hidden units, nh ), the |V| factor can be reduced to as low as log |V|. Bengio (2002) and Morin and Bengio (2005) introduced this factorized approach to the context of neural language models. One can think of this hierarchy as building categories of words, then categories of categories of words, then categories of categories of categories of words, etc. These nested categories form a tree, with words at the leaves. In a balanced tree, the tree has depth O(log |V|). The probability of a choosing a word is given by the product of the probabilities of choosing the branch leading to that word at every node on a path from the root of the tree to the leaf containing the word. Fig. 12.4 illustrates a simple example. Mnih and Hinton (2009) also describe how to use multiple paths to identify a single word in order to better model words that have multiple meanings. Computing the probability of a word then involves summation over all of the paths that lead to that word. To predict the conditional probabilities required at each node of the tree, we typically use a logistic regression model at each node of the tree, and provide the same context C as input to all of these models. Because the correct output is encoded in the training set, we can use supervised learning to train the logistic regression models. This is typically done using a standard cross-entropy loss, corresponding to maximizing the log-likelihood of the correct sequence of decisions. Because the output log-likelihood can be computed eﬃciently (as low as log |V| rather than |V| ), its gradients may also be computed eﬃciently. This includes not only the gradient with respect to the output parameters but also the gradients with respect to the hidden layer activations. It is possible but usually not practical to optimize the tree structure to minimize the expected number of computations. Tools from information theory specify how to choose the optimal binary code given the relative frequencies of the words. To do so, we could structure the tree so that the number of bits associated with a word is approximately equal to the logarithm of the frequency of that word. However, in practice, the computational savings are typically not worth the eﬀort because the computation of the output probabilities is only one part of the total computation in the neural language model. For example, suppose there are l fully connected hidden layers of width nh . Let nb be the weighted average of the number of bits 469

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(0)

(1)

(0,0)

(0,1)

(1,0)

(1,1)

w0

w1

w2

w3

w4

w5

w6

w7

(0,0,0)

(0,0,1)

(0,1,0)

(0,1,1)

(1,0,0)

(1,0,1)

(1,1,0)

(1,1,1)

Figure 12.4: Illustration of a simple hierarchy of word categories, with 8 words w 0, . . . , w7 organized into a three level hierarchy. The leaves of the tree represent actual speciﬁc words. Internal nodes represent groups of words. Any node can be indexed by the sequence of binary decisions (0=left, 1=right) to reach the node from the root. Super-class (0) contains the classes (0, 0) and (0 , 1), which respectively contain the sets of words {w0 , w1 } and {w 2, w3 } , and similarly super-class (1) contains the classes (1,0) and (1, 1), which respectively contain the words (w 4, w 5 ) and ( w6, w 7). If the tree is suﬃciently balanced, the maximum depth (number of binary decisions) is on the order of the logarithm of the number of words |V| : the choice of one out of |V| words can be obtained by doing O(log |V|) operations (one for each of the nodes on the path from the root). In this example, computing the probability of a word y can be done by multiplying three probabilities, associated with the binary decisions to move left or right at each node on the path from the root to a node y . Let bi(y ) be the i-th binary decision when traversing the tree towards the value y . The probability of sampling an output y decomposes into a product of conditional probabilities, using the chain rule for conditional probabilities, with each node indexed by the preﬁx of these bits. For example, node (1, 0) corresponds to the preﬁx (b0 (w4 ) = 1, b1 (w4 ) = 0), and the probability of w4 can be decomposed as follows: P (y = w 4) = P (b0 = 1, b1 = 0, b2 = 0) = P (b0 = 1)P (b 1 = 0 | b0 = 1)P (b 2 = 0 | b0 = 1, b1 = 0).

470

(12.12) (12.13)

CHAPTER 12. APPLICATIONS

required to identify a word, with the weighting given by the frequency of these words. In this example, the number of operations needed to compute the hidden activations grows as as O(ln2h ) while the output computations grow as O(nh nb ). As long as nb ≤ ln h, we can reduce computation more by shrinking nh than by shrinking nb . Indeed, nb is often small. Because the size of the vocabulary rarely exceeds a million words and log2 (106) ≈ 20, it is possible to reduce nb to about 20, but n h is often much larger, around 103 or more. Rather than carefully optimizing a tree with a branching factor of 2, one can instead deﬁne a tree with depth two and a branching factor of |V|. Such a tree corresponds to simply deﬁning a set of mutually exclusive word classes. The simple approach based on a tree of depth two captures most of the computational beneﬁt of the hierarchical strategy. One question that remains somewhat open is how to best deﬁne these word classes, or how to deﬁne the word hierarchy in general. Early work used existing hierarchies (Morin and Bengio, 2005) but the hierarchy can also be learned, ideally jointly with the neural language model. Learning the hierarchy is diﬃcult. An exact optimization of the log-likelihood appears intractable because the choice of a word hierarchy is a discrete one, not amenable to gradient-based optimization. However, one could use discrete optimization to approximately optimize the partition of words into word classes. An important advantage of the hierarchical softmax is that it brings computational beneﬁts both at training time and at test time, if at test time we want to compute the probability of speciﬁc words. Of course, computing the probability of all |V| words will remain expensive even with the hierarchical softmax. Another important operation is selecting the most likely word in a given context. Unfortunately the tree structure does not provide an eﬃcient and exact solution to this problem. A disadvantage is that in practice the hierarchical softmax tends to give worse test results than sampling-based methods we will describe next. This may be due to a poor choice of word classes. 12.4.3.3

Importance Sampling

One way to speed up the training of neural language models is to avoid explicitly computing the contribution of the gradient from all of the words that do not appear in the next position. Every incorrect word should have low probability under the model. It can be computationally costly to enumerate all of these words. Instead, it is possible to sample only a subset of the words. Using the notation introduced

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in Eq. 12.8, the gradient can be written as follows: ∂ log softmaxy (a) ∂ log P (y | C ) = ∂θ ∂θ ∂ eay = log a i ∂θ ie ∂ = (a y − log ea i ) ∂θ i ∂ay ∂ai = − P (y = i | C ) ∂θ ∂θ i

(12.14) (12.15) (12.16) (12.17)

where a is the vector of pre-softmax activations (or scores), with one element per word. The ﬁrst term is the positive phase term (pushing ay up) while the second term is the negative phase term (pushing ai down for all i, with weight P( i | C ). Since the negative phase term is an expectation, we can estimate it with a Monte Carlo sample. However, that would require sampling from the model itself. Sampling from the model requires computing P (i | C ) for all i in the vocabulary, which is precisely what we are trying to avoid. Instead of sampling from the model, one can sample from another distribution, called the proposal distribution (denoted q ), and use appropriate weights to correct for the bias introduced by sampling from the wrong distribution (Bengio and Sénécal, 2003; Bengio and Sénécal, 2008). This is an application of a more general technique called importance sampling, which will be described in more detail in Sec. 17.2. Unfortunately, even exact importance sampling is not eﬃcient because it requires computing weights pi /qi , where p i = P (i | C ), which can only be computed if all the scores ai are computed. The solution adopted for this application is called biased importance sampling, where the importance weights are normalized to sum to 1. When negative word ni is sampled, the associated gradient is weighted by p n /qn i wi = N i . j=1 pnj /qn j

(12.18)

These weights are used to give the appropriate importance to the m negative samples from q used to form the estimated negative phase contribution to the gradient: |V| m ∂a i 1 ∂ani wi . (12.19) P (i | C ) ≈ ∂θ m ∂θ i=1 i=1

A unigram or a bigram distribution works well as the proposal distribution q . It is easy to estimate the parameters of such a distribution from data. After estimating the parameters, it is also possible to sample from such a distribution very eﬃciently. 472

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Importance sampling is not only useful for speeding up models with large softmax outputs. More generally, it is useful for accelerating training with large sparse output layers, where the output is a sparse vector rather than a 1-of- n choice. An example is a bag of words. A bag of words is a sparse vector v where vi indicates the presence or absence of word i from the vocabulary in the document. Alternately, vi can indicate the number of times that word i appears. Machine learning models that emit such sparse vectors can be expensive to train for a variety of reasons. Early in learning, the model may not actually choose to make the output truly sparse. Moreover, the loss function we use for training might most naturally be described in terms of comparing every element of the output to every element of the target. This means that it is not always clear that there is a computational beneﬁt to using sparse outputs, because the model may choose to make the majority of the output non-zero and all of these non-zero values need to be compared to the corresponding training target, even if the training target is zero. Dauphin et al. (2011) demonstrated that such models can be accelerated using importance sampling. The eﬃcient algorithm minimizes the loss reconstruction for the “positive words” (those that are non-zero in the target) and an equal number of “negative words.” The negative words are chosen randomly, using a heuristic to sample words that are more likely to be mistaken. The bias introduced by this heuristic oversampling can then be corrected using importance weights. In all of these cases, the computational complexity of gradient estimation for the output layer is reduced to be proportional to the number of negative samples rather than proportional to the size of the output vector. 12.4.3.4

Noise-Contrastive Estimation and Ranking Loss

Other approaches based on sampling have been proposed to reduce the computational cost of training neural language models with large vocabularies. An early example is the ranking loss proposed by Collobert and Weston (2008a), which views the output of the neural language model for each word as a score and tries to make the score of the correct word ay be ranked high in comparison to the other scores ai . The ranking loss proposed then is L= (12.20) max(0, 1 − ay + ai ). i

The gradient is zero for the i-th term if the score of the observed word, a y, is greater than the score of the negative word ai by a margin of 1. One issue with this criterion is that it does not provide estimated conditional probabilities, which 473

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are useful in some applications, including speech recognition and text generation (including conditional text generation tasks such as translation). A more recently used training objective for neural language model is noisecontrastive estimation, which is introduced in Sec. 18.6. This approach has been successfully applied to neural language models (Mnih and Teh, 2012; Mnih and Kavukcuoglu, 2013).

12.4.4

Combining Neural Language Models with n-grams

A major advantage of n-gram models over neural networks is that n-gram models achieve high model capacity (by storing the frequencies of very many tuples) while requiring very little computation to process an example (by looking up only a few tuples that match the current context). If we use hash tables or trees to access the counts, the computation used for n-grams is almost independent of capacity. In comparison, doubling a neural network’s number of parameters typically also roughly doubles its computation time. Exceptions include models that avoid using all parameters on each pass. Embedding layers index only a single embedding in each pass, so we can increase the vocabulary size without increasing the computation time per example. Some other models, such as tiled convolutional networks, can add parameters while reducing the degree of parameter sharing in order to maintain the same amount of computation. However, typical neural network layers based on matrix multiplication use an amount of computation proportional to the number of parameters. One easy way to add capacity is thus to combine both approaches in an ensemble consisting of a neural language model and an n-gram language model (Bengio et al., 2001, 2003). As with any ensemble, this technique can reduce test error if the ensemble members make independent mistakes. The ﬁeld of ensemble learning provides many ways of combining the ensemble members’ predictions, including uniform weighting and weights chosen on a validation set. Mikolov et al. (2011a) extended the ensemble to include not just two models but a large array of models. It is also possible to pair a neural network with a maximum entropy model and train both jointly (Mikolov et al., 2011b). This approach can be viewed as training a neural network with an extra set of inputs that are connected directly to the output, and not connected to any other part of the model. The extra inputs are indicators for the presence of particular n-grams in the input context, so these variables are very high-dimensional and very sparse. The increase in model capacity is huge—the new portion of the architecture contains up to |sV |n parameters—but the amount of added computation needed to process an input is minimal because the extra inputs are very sparse. 474

CHAPTER 12. APPLICATIONS

12.4.5

Neural Machine Translation

Machine translation is the task of reading a sentence in one natural language and emitting a sentence with the equivalent meaning in another language. Machine translation systems often involve many components. At a high level, there is often one component that proposes many candidate translations. Many of these translations will not be grammatical due to diﬀerences between the languages. For example, many languages put adjectives after nouns, so when translated to English directly they yield phrases such as “apple red.” The proposal mechanism suggests many variants of the suggested translation, ideally including “red apple.” A second component of the translation system, a language model, evaluates the proposed translations, and can score “red apple” as better than “apple red.” The earliest use of neural networks for machine translation was to upgrade the language model of a translation system by using a neural language model (Schwenk et al., 2006; Schwenk, 2010). Previously, most machine translation systems had used an n-gram model for this component. The n-gram based models used for machine translation include not just traditional back-oﬀ n-gram models (Jelinek and Mercer, 1980; Katz, 1987; Chen and Goodman, 1999) but also maximum entropy language models (Berger et al., 1996), in which an aﬃne-softmax layer predicts the next word given the presence of frequent n-grams in the context. Traditional language models simply report the probability of a natural language sentence. Because machine translation involves producing an output sentence given an input sentence, it makes sense to extend the natural language model to be conditional. As described in Sec. 6.2.1.1, it is straightforward to extend a model that deﬁnes a marginal distribution over some variable to deﬁne a conditional distribution over that variable given a context C, where C might be a single variable or a list of variables. Devlin et al. (2014) beat the state-of-the-art in some statistical machine translation benchmarks by using an MLP to score a phrase t 1, t2 , . . . , tk in the target language given a phrase s1 , s2 , . . . , sn in the source language. The MLP estimates P (t1 , t2 , . . . , tk | s1, s2 , . . . , sn ). The estimate formed by this MLP replaces the estimate provided by conditional n-gram models. A drawback of the MLP-based approach is that it requires the sequences to be preprocessed to be of ﬁxed length. To make the translation more ﬂexible, we would like to use a model that can accommodate variable length inputs and variable length outputs. An RNN provides this ability. Sec. 10.2.4 describes several ways of constructing an RNN that represents a conditional distribution over a sequence given some input, and Sec. 10.4 describes how to accomplish this conditioning when the input is a sequence. In all cases, one model ﬁrst reads the input sequence and emits a data structure that summarizes the input sequence. We call this 475

CHAPTER 12. APPLICATIONS

Output object (English sentence) Decoder

Intermediate, semantic representation

Encoder

Source object (French sentence or image)

Figure 12.5: The encoder-decoder architecture to map back and forth between a surface representation (such as a sequence of words or an image) and a semantic representation. By using the output of an encoder of data from one modality (such as the encoder mapping from French sentences to hidden representations capturing the meaning of sentences) as the input to a decoder for another modality (such as the decoder mapping from hidden representations capturing the meaning of sentences to English), we can train systems that translate from one modality to another. This idea has been applied successfully not just to machine translation but also to caption generation from images.

summary the “context” C. The context C may be a list of vectors, or it may be a vector or tensor. The model that reads the input to produce C may be an RNN (Cho et al., 2014a; Sutskever et al., 2014; Jean et al., 2014) or a convolutional network (Kalchbrenner and Blunsom, 2013). A second model, usually an RNN, then reads the context C and generates a sentence in the target language. This general idea of an encoder-decoder framework for machine translation is illustrated in Fig. 12.5. In order to generate an entire sentence conditioned on the source sentence, the model must have a way to represent the entire source sentence. Earlier models were only able to represent individual words or phrases. From a representation learning point of view, it can be useful to learn a representation in which sentences that have the same meaning have similar representations regardless of whether they were written in the source language or the target language. This strategy was explored ﬁrst using a combination of convolutions and RNNs (Kalchbrenner and Blunsom, 2013). Later work introduced the use of an RNN for scoring proposed translations (Cho et al., 2014a) and for generating translated sentences (Sutskever et al., 2014). Jean et al. (2014) scaled these models to larger vocabularies.

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12.4.5.1

Using an Attention Mechanism and Aligning Pieces of Data

c +

α (t−1)

α (t)

× h(t−1)

α(t+1) ×

h(t)

× h(t+1)

Figure 12.6: A modern attention mechanism, as introduced by Bahdanau et al. (2015), is essentially a weighted average. A context vector c is formed by taking a weighted average of feature vectors h (t) with weights α (t) . In some applications, the feature vectors h are hidden units of a neural network, but they may also be raw input to the model. The weights α (t) are produced by the model itself. They are usually values in the interval [0, 1] and are intended to concentrate around just one h(t) so that the weighted average approximates reading that one speciﬁc time step precisely. The weights α(t) are usually produced by applying a softmax function to relevance scores emitted by another portion of the model. The attention mechanism is more expensive computationally than directly indexing the desired h(t) , but direct indexing cannot be trained with gradient descent. The attention mechanism based on weighted averages is a smooth, diﬀerentiable approximation that can be trained with existing optimization algorithms.

Using a ﬁxed-size representation to capture all the semantic details of a very long sentence of say 60 words is very diﬃcult. It can be achieved by training a suﬃciently large RNN well enough and for long enough, as demonstrated by Cho et al. (2014a) and Sutskever et al. (2014). However, a more eﬃcient approach is to read the whole sentence or paragraph (to get the context and the gist of what is being expressed), then produce the translated words one at a time, each time focusing on a diﬀerent part of the input sentence in order to gather the semantic details that are required to produce the next output word. That is exactly the idea that Bahdanau et al. (2015) ﬁrst introduced. The attention mechanism used to focus on speciﬁc parts of the input sequence at each time step is illustrated in Fig. 12.6. We can think of an attention-based system as having three components: 477

CHAPTER 12. APPLICATIONS

1. A process that “ reads” raw data (such as source words in a source sentence), and converts them into distributed representations, with one feature vector associated with each word position. 2. A list of feature vectors storing the output of the reader. This can be understood as a “ memory” containing a sequence of facts, which can be retrieved later, not necessarily in the same order, without having to visit all of them. 3. A process that “ exploits” the content of the memory to sequentially perform a task, at each time step having the ability put attention on the content of one memory element (or a few, with a diﬀerent weight). The third component generates the translated sentence. When words in a sentence written in one language are aligned with corresponding words in a translated sentence in another language, it becomes possible to relate the corresponding word embeddings. Earlier work showed that one could learn a kind of translation matrix relating the word embeddings in one language with the word embeddings in another (Kočiský et al., 2014), yielding lower alignment error rates than traditional approaches based on the frequency counts in the phrase table. There is even earlier work on learning cross-lingual word vectors (Klementiev et al., 2012). Many extensions to this approach are possible. For example, more eﬃcient cross-lingual alignment (Gouws et al., 2014) allows training on larger datasets.

12.4.6

Historical Perspective

The idea of distributed representations for symbols was introduced by Rumelhart et al. (1986a) in one of the ﬁrst explorations of back-propagation, with symbols corresponding to the identity of family members and the neural network capturing the relationships between family members, with training examples forming triplets such as (Colin, Mother, Victoria). The ﬁrst layer of the neural network learned a representation of each family member. For example, the features for Colin might represent which family tree Colin was in, what branch of that tree he was in, what generation he was from, etc. One can think of the neural network as computing learned rules relating these attributes together in order to obtain the desired predictions. The model can then make predictions such as inferring who is the mother of Colin. The idea of forming an embedding for a symbol was extended to the idea of an embedding for a word by Deerwester et al. (1990). These embeddings were learned using the SVD. Later, embeddings would be learned by neural networks. 478

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The history of natural language processing is marked by transitions in the popularity of diﬀerent ways of representing the input to the model. Following this early work on symbols or words, some of the earliest applications of neural networks to NLP (Miikkulainen and Dyer, 1991; Schmidhuber, 1996) represented the input as a sequence of characters. Bengio et al. (2001) returned the focus to modeling words and introduced neural language models, which produce interpretable word embeddings. These neural models have scaled up from deﬁning representations of a small set of symbols in the 1980s to millions of words (including proper nouns and misspellings) in modern applications. This computational scaling eﬀort led to the invention of the techniques described above in Sec. 12.4.3. Initially, the use of words as the fundamental units of language models yielded improved language modeling performance (Bengio et al., 2001). To this day, new techniques continually push both character-based models (Sutskever et al., 2011) and word-based models forward, with recent work (Gillick et al., 2015) even modeling individual bytes of Unicode characters. The ideas behind neural language models have been extended into several natural language processing applications, such as parsing (Henderson, 2003, 2004; Collobert, 2011), part-of-speech tagging, semantic role labeling, chunking, etc, sometimes using a single multi-task learning architecture (Collobert and Weston, 2008a; Collobert et al., 2011a) in which the word embeddings are shared across tasks. Two-dimensional visualizations of embeddings became a popular tool for analyzing language models following the development of the t-SNE dimensionality reduction algorithm (van der Maaten and Hinton, 2008) and its high-proﬁle application to visualization word embeddings by Joseph Turian in 2009.

12.5

Other Applications

In this section we cover a few other types of applications of deep learning that are diﬀerent from the standard object recognition, speech recognition and natural language processing tasks discussed above. Part III of this book will expand that scope even further to include tasks requiring the ability to generate rich high-dimensional samples (unlike “the next word,” in language models).

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12.5.1

Recommender Systems

One of the major families of applications of machine learning in the information technology sector is the ability to make recommendations of items to potential users or customers. Two major types of applications can be distinguished: online advertising and item recommendations (often these recommendations are still for the purpose of selling a product). Both rely on predicting the association between a user and an item, either to predict the probability of some action (the user buying the product, or some proxy for this action) or the expected gain (which may depend on the value of the product) if an ad is shown or a recommendation is made regarding that product to that user. The internet is currently ﬁnanced in great part by various forms of online advertising. There are major parts of the economy that rely on online shopping. Companies including Amazon and eBay use machine learning, including deep learning, for their product recommendations. Sometimes, the items are not products that are actually for sale. Examples include selecting posts to display on social network news feeds, recommending movies to watch, recommending jokes, recommending advice from experts, matching players for video games, or matching people in dating services. Often, this association problem is handled like a supervised learning problem: given some information about the item and about the user, predict the proxy of interest (user clicks on ad, user enters a rating, user clicks on a “like” button, user buys product, user spends some amount of money on the product, user spends time visiting a page for the product, etc). This often ends up being either a regression problem (predicting some conditional expected value) or a probabilistic classiﬁcation problem (predicting the conditional probability of some discrete event). The early work on recommender systems relied on minimal information as inputs for these predictions: the user ID and the item ID. In this context, the only way to generalize is to rely on the similarity between the patterns of values of the target variable for diﬀerent users or for diﬀerent items. Suppose that user 1 and user 2 both like items A, B and C. From this, we may infer that user 1 and user 2 have similar tastes. If user 1 likes item D, then this should be a strong cue that user 2 will also like D. Algorithms based on this principle come under the name of collaborative ﬁltering. Both non-parametric approaches (such as nearest-neighbor methods based on the estimated similarity between patterns of preferences) and parametric methods are possible. Parametric methods often rely on learning a distributed representation (also called an embedding) for each user and for each item. Bilinear prediction of the target variable (such as a rating) is a simple parametric method that is highly successful and often found as a component of 480

CHAPTER 12. APPLICATIONS

state-of-the-art systems. The prediction is obtained by the dot product between the user embedding and the item embedding (possibly corrected by constants that ˆ be the matrix containing depend only on either the user ID or the item ID). Let R our predictions, A a matrix with user embeddings in its rows and B a matrix with item embeddings in its columns. Let b and c be vectors that contain respectively a kind of bias for each user (representing how grumpy or positive that user is in general) and for each item (representing its general popularity). The bilinear prediction is thus obtained as follows: ˆu,i = bu + c i + R A u,j Bj,i . (12.21) j

Typically one wants to minimize the squared error between predicted ratings ˆu,i and actual ratings Ru,i . User embeddings and item embeddings can then be R conveniently visualized when they are ﬁrst reduced to a low dimension (two or three), or they can be used to compare users or items against each other, just like word embeddings. One way to obtain these embeddings is by performing a singular value decomposition of the matrix R of actual targets (such as ratings). This corresponds to factorizing R = U DV (or a normalized variant) into the product of two factors, the lower rank matrices A = U D and B = V . One problem with the SVD is that it treats the missing entries in an arbitrary way, as if they corresponded to a target value of 0. Instead we would like to avoid paying any cost for the predictions made on missing entries. Fortunately, the sum of squared errors on the observed ratings can also be easily minimized by gradient-based optimization. The SVD and the bilinear prediction of Eq. 12.21 both performed very well in the competition for the Netﬂix prize (Bennett and Lanning, 2007), aiming at predicting ratings for ﬁlms, based only on previous ratings by a large set of anonymous users. Many machine learning experts participated in this competition, which took place between 2006 and 2009. It raised the level of research in recommender systems using advanced machine learning and yielded improvements in recommender systems. Even though it did not win by itself, the simple bilinear prediction or SVD was a component of the ensemble models presented by most of the competitors, including the winners (Töscher et al., 2009; Koren, 2009). Beyond these bilinear models with distributed representations, one of the ﬁrst uses of neural networks for collaborative ﬁltering is based on the RBM undirected probabilistic model (Salakhutdinov et al., 2007). RBMs were an important element of the ensemble of methods that won the Netﬂix competition (Töscher et al., 2009; Koren, 2009). More advanced variants on the idea of factorizing the ratings matrix have also been explored in the neural networks community (Salakhutdinov and 481

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Mnih, 2008). However, there is a basic limitation of collaborative ﬁltering systems: when a new item or a new user is introduced, its lack of rating history means that there is no way to evaluate its similarity with other items or users (respectively), or the degree of association between, say, that new user and existing items. This is called the problem of cold-start recommendations. A general way of solving the cold-start recommendation problem is to introduce extra information about the individual users and items. For example, this extra information could be user proﬁle information or features of each item. Systems that use such information are called content-based recommender systems. The mapping from a rich set of user features or item features to an embedding can be learned through a deep learning architecture (Huang et al., 2013; Elkahky et al., 2015). Specialized deep learning architectures such as convolutional networks have also been applied to learn to extract features from rich content such as from musical audio tracks, for music recommendation (van den Oörd et al., 2013). In that work, the convolutional net takes acoustic features as input and computes an embedding for the associated song. The dot product between this song embedding and the embedding for a user is then used to predict whether a user will listen to the song. 12.5.1.1

Exploration Versus Exploitation

When making recommendations to users, an issue arises that goes beyond ordinary supervised learning and into the realm of reinforcement learning. Many recommendation problems are most accurately described theoretically as contextual bandits (Langford and Zhang, 2008; Lu et al., 2010). The issue is that when we use the recommendation system to collect data, we get a biased and incomplete view of the preferences of users: we only see the responses of users to the items they were recommended and not to the other items. In addition, in some cases we may not get any information on users for whom no recommendation has been made (for example, with ad auctions, it may be that the price proposed for an ad was below a minimum price threshold, or does not win the auction, so the ad is not shown at all). More importantly, we get no information about what outcome would have resulted from recommending any of the other items. This would be like training a classiﬁer by picking one class yˆ for each training example x (typically the class with the highest probability according to the model) and then only getting as feedback whether this was the correct class or not. Clearly, each example conveys less information than in the supervised case where the true label y is directly accessible, so more examples are necessary. Worse, if we are not careful, we could end up with a system that continues picking the wrong decisions even as more 482

CHAPTER 12. APPLICATIONS

and more data is collected, because the correct decision initially had a very low probability: until the learner picks that correct decision, it does not learn about the correct decision. This is similar to the situation in reinforcement learning where only the reward for the selected action is observed. In general, reinforcement learning can involve a sequence of many actions and many rewards. The bandits scenario is a special case of reinforcement learning, in which the learner takes only a single action and receives a single reward. The bandit problem is easier in the sense that the learner knows which reward is associated with which action. In the general reinforcement learning scenario, a high reward or a low reward might have been caused by a recent action or by an action in the distant past. The term contextual bandits refers to the case where the action is taken in the context of some input variable that can inform the decision. For example, we at least know the user identity, and we want to pick an item. The mapping from context to action is also called a policy. The feedback loop between the learner and the data distribution (which now depends on the actions of the learner) is a central research issue in the reinforcement learning and bandits literature. Reinforcement learning requires choosing a tradeoﬀ between exploration and exploitation. Exploitation refers to taking actions that come from the current, best version of the learned policy—actions that we know will achieve a high reward. Exploration refers to taking actions speciﬁcally in order to obtain more training data. If we know that given context x, action a gives us a reward of 1, we do not know whether that is the best possible reward. We may want to exploit our current policy and continue taking action a in order to be relatively sure of obtaining a reward of 1. However, we may also want to explore by trying action a. We do not know what will happen if we try action a . We hope to get a reward of 2, but we run the risk of getting a reward of 0. Either way, we at least gain some knowledge. Exploration can be implemented in many ways, ranging from occasionally taking random actions intended to cover the entire space of possible actions, to model-based approaches that compute a choice of action based on its expected reward and the model’s amount of uncertainty about that reward. Many factors determine the extent to which we prefer exploration or exploitation. One of the most prominent factors is the time scale we are interested in. If the agent has only a short amount of time to accrue reward, then we prefer more exploitation. If the agent has a long time to accrue reward, then we begin with more exploration so that future actions can be planned more eﬀectively with more knowledge. As time progresses and our learned policy improves, we move toward more exploitation. Supervised learning has no tradeoﬀ between exploration and exploitation 483

CHAPTER 12. APPLICATIONS

because the supervision signal always speciﬁes which output is correct for each input. There is no need to try out diﬀerent outputs to determine if one is better than the model’s current output—we always know that the label is the best output. Another diﬃculty arising in the context of reinforcement learning, besides the exploration-exploitation trade-oﬀ, is the diﬃculty of evaluating and comparing diﬀerent policies. Reinforcement learning involves interaction between the learner and the environment. This feedback loop means that it is not straightforward to evaluate the learner’s performance using a ﬁxed set of test set input values. The policy itself determines which inputs will be seen. Dudik et al. (2011) present techniques for evaluating contextual bandits.

12.5.2

Knowledge Representation, Reasoning and Question Answering

Deep learning approaches have been very successful in language modeling, machine translation and natural language processing due to the use of embeddings for symbols (Rumelhart et al., 1986a) and words (Deerwester et al., 1990; Bengio et al., 2001). These embeddings represent semantic knowledge about individual words and concepts. A research frontier is to develop embeddings for phrases and for relations between words and facts. Search engines already use machine learning for this purpose but much more remains to be done to improve these more advanced representations. 12.5.2.1

Knowledge, Relations and Question Answering

indexRelations One interesting research direction is determining how distributed representations can be trained to capture the relations between two entities. These relations allow us to formalize facts about objects and how objects interact with each other. In mathematics, a binary relation is a set of ordered pairs of objects. Pairs that are in the set are said to have the relation while those who are not in the set do not. For example, we can deﬁne the relation “is less than” on the set of entities {1, 2, 3} by deﬁning the set of ordered pairs S = {(1 ,2), (1, 3), (2, 3)}. Once this relation is deﬁned, we can use it like a verb. Because (1, 2) ∈ S, we say that 1 is less than 2. Because (2,1) ∈ S , we can not say that 2 is less than 1. Of course, the entities that are related to one another need not be numbers. We could deﬁne a relation is_a_type_of containing tuples like (dog, mammal). In the context of AI, we think of a relation as a sentence in a syntactically 484

CHAPTER 12. APPLICATIONS

simple and highly structured language. The relation plays the role of a verb, while two arguments to the relation play the role of its subject and object. These sentences take the form of a triplet of tokens (subject, verb, object)

(12.22)

(entity i, relationj, entity k).

(12.23)

with values We can also deﬁne an attribute, a concept analogous to a relation, but taking only one argument: (entityi, attribute j). (12.24) For example, we could deﬁne the has_fur attribute, and apply it to entities like dog. Many applications require representing relations and reasoning about them. How should we best do this within the context of neural networks? Machine learning models of course require training data. We can infer relations between entities from training datasets consisting of unstructured natural language. There are also structured databases that identify relations explicitly. A common structure for these databases is the relational database, which stores this same kind of information, albeit not formatted as three token sentences. When a database is intended to convey commonsense knowledge about everyday life or expert knowledge about an application area to an artiﬁcial intelligence system, we call the database a knowledge base. Knowledge bases range from general ones like Freebase , OpenCyc, WordNet, or Wikibase, 1 etc. to more specialized knowledge bases, like GeneOntology.2 Representations for entities and relations can be learned by considering each triplet in a knowledge base as a training example and maximizing a training objective that captures their joint distribution (Bordes et al., 2013a). In addition to training data, we also need to deﬁne a model family to train. A common approach is to extend neural language models to model entities and relations. Neural language models learn a vector that provides a distributed representation of each word. They also learn about interactions between words, such as which word is likely to come after a sequence of words, by learning functions of these vectors. We can extend this approach to entities and relations by learning an embedding vector for each relation. In fact, the parallel between modeling 1

Respectively available from these web sites: freebase.com, cyc.com/opencyc, wordnet. princeton.edu, wikiba.se 2 geneontology.org 485

CHAPTER 12. APPLICATIONS

language and modeling knowledge encoded as relations is so close that researchers have trained representations of such entities by using both knowledge bases and natural language sentences (Bordes et al., 2011, 2012; Wang et al., 2014a) or combining data from multiple relational databases (Bordes et al., 2013b). Many possibilities exist for the particular parametrization associated with such a model. Early work on learning about relations between entities (Paccanaro and Hinton, 2000) posited highly constrained parametric forms (“linear relational embeddings”), often using a diﬀerent form of representation for the relation than for the entities. For example, Paccanaro and Hinton (2000) and Bordes et al. (2011) used vectors for entities and matrices for relations, with the idea that a relation acts like an operator on entities. Alternatively, relations can be considered as any other entity (Bordes et al., 2012), allowing us to make statements about relations, but more ﬂexibility is put in the machinery that combines them in order to model their joint distribution. A practical short-term application of such models is link prediction: predicting missing arcs in the knowledge graph. This is a form of generalization to new facts, based on old facts. Most of the knowledge bases that currently exist have been constructed through manual labor, which tends to leave many and probably the majority of true relations absent from the knowledge base. See Wang et al. (2014b), Lin et al. (2015) and Garcia-Duran et al. (2015) for examples of such an application. Evaluating the performance of a model on a link prediction task is diﬃcult because we have only a dataset of positive examples (facts that are known to be true). If the model proposes a fact that is not in the dataset, we are unsure whether the model has made a mistake or discovered a new, previously unknown fact. The metrics are thus somewhat imprecise and are based on testing how the model ranks a held-out of set of known true positive facts compared to other facts that are less likely to be true. A common way to construct interesting examples that are probably negative (facts that are probably false) is to begin with a true fact and create corrupted versions of that fact, for example by replacing one entity in the relation with a diﬀerent entity selected at random. The popular precision at 10% metric counts how many times the model ranks a “correct” fact among the top 10% of all corrupted versions of that fact. Another application of knowledge bases and distributed representations for them is word-sense disambiguation (Navigli and Velardi, 2005; Bordes et al., 2012), which is the task of deciding which of the senses of a word is the appropriate one, in some context. Eventually, knowledge of relations combined with a reasoning process and understanding of natural language could allow us to build a general question 486

CHAPTER 12. APPLICATIONS

answering system. A general question answering system must be able to process input information and remember important facts, organized in a way that enables it to retrieve and reason about them later. This remains a diﬃcult open problem which can only be solved in restricted “toy” environments. Currently, the best approach to remembering and retrieving speciﬁc declarative facts is to use an explicit memory mechanism, as described in Sec. 10.12. Memory networks were ﬁrst proposed to solve a toy question answering task (Weston et al., 2014). Kumar et al. (2015) have proposed an extension that uses GRU recurrent nets to read the input into the memory and to produce the answer given the contents of the memory. Deep learning has been applied to many other applications besides the ones described here, and will surely be applied to even more after this writing. It would be impossible to describe anything remotely resembling a comprehensive coverage of such a topic. This survey provides a representative sample of what is possible as of this writing. This concludes Part II, which has described modern practices involving deep networks, comprising all of the most successful methods. Generally speaking, these methods involve using the gradient of a cost function to ﬁnd the parameters of a model that approximates some desired function. With enough training data, this approach is extremely powerful. We now turn to Part III, in which we step into the territory of research—methods that are designed to work with less training data or to perform a greater variety of tasks, where the challenges are more diﬃcult and not as close to being solved as the situations we have described so far.

487

Part III

Deep Learning Research

488

This part of the book describes the more ambitious and advanced approaches to deep learning, currently pursued by the research community. In the previous parts of the book, we have shown how to solve supervised learning problems—how to learn to map one vector to another, given enough examples of the mapping. Not all problems we might want to solve fall into this category. We may wish to generate new examples, or determine how likely some point is, or handle missing values and take advantage of a large set of unlabeled examples or examples from related tasks. A shortcoming of the current state of the art for industrial applications is that our learning algorithms require large amounts of supervised data to achieve good accuracy. In this part of the book, we discuss some of the speculative approaches to reducing the amount of labeled data necessary for existing models to work well and be applicable across a broader range of tasks. Accomplishing these goals usually requires some form of unsupervised or semi-supervised learning. Many deep learning algorithms have been designed to tackle unsupervised learning problems, but none have truly solved the problem in the same way that deep learning has largely solved the supervised learning problem for a wide variety of tasks. In this part of the book, we describe the existing approaches to unsupervised learning and some of the popular thought about how we can make progress in this ﬁeld. A central cause of the diﬃculties with unsupervised learning is the high dimensionality of the random variables being modeled. This brings two distinct challenges: a statistical challenge and a computational challenge. The statistical challenge regards generalization: the number of conﬁgurations we may want to distinguish can grow exponentially with the number of dimensions of interest, and this quickly becomes much larger than the number of examples one can possibly have (or use with bounded computational resources). The computational challenge associated with high-dimensional distributions arises because many algorithms for learning or using a trained model (especially those based on estimating an explicit probability function) involve intractable computations that grow exponentially with the number of dimensions. With probabilistic models, this computational challenge arises from the need to perform intractable inference or simply from the need to normalize the distribution. • Intractable inference: inference is discussed mostly in Chapter 19. It regards the question of guessing the probable values of some variables a, given other variables b, with respect to a model that captures the joint 489

distribution between a, b and c. In order to even compute such conditional probabilities one needs to sum over the values of the variables c, as well as compute a normalization constant which sums over the values of a and c. • Intractable normalization constants (the partition function): the partition function is discussed mostly in Chapter 18. Normalizing constants of probability functions come up in inference (above) as well as in learning. Many probabilistic models involve such a normalizing constant. Unfortunately, learning such a model often requires computing the gradient of the logarithm of the partition function with respect to the model parameters. That computation is generally as intractable as computing the partition function itself. Monte Carlo Markov chain (MCMC) methods (Chapter 17) are often used to deal with the partition function (computing it or its gradient). Unfortunately, MCMC methods suﬀer when the modes of the model distribution are numerous and well-separated, especially in high-dimensional spaces (Sec. 17.5). One way to confront these intractable computations is to approximate them, and many approaches have been proposed as discussed in this third part of the book. Another interesting way, also discussed here, would be to avoid these intractable computations altogether by design, and methods that do not require such computations are thus very appealing. Several generative models have been proposed in recent years, with that motivation. A wide variety of contemporary approaches to generative modeling are discussed in Chapter 20. Part III is the most important for a researcher—someone who wants to understand the breadth of perspectives that have been brought to the ﬁeld of deep learning, and push the ﬁeld forward towards true artiﬁcial intelligence.

490

Chapter 13

Linear Factor Models Many of the research frontiers in deep learning involve building a probabilistic model of the input, pmodel (x). Such a model can, in principle, use probabilistic inference to predict any of the variables in its environment given any of the other variables. Many of these models also have latent variables h, with pmodel (x) = Eh pmodel (x | h). These latent variables provide another means of representing the data. Distributed representations based on latent variables can obtain all of the advantages of representation learning that we have seen with deep feedforward and recurrent networks. In this chapter, we describe some of the simplest probabilistic models with latent variables: linear factor models. These models are sometimes used as building blocks of mixture models (Hinton et al., 1995a; Ghahramani and Hinton, 1996; Roweis et al., 2002) or larger, deep probabilistic models (Tang et al., 2012). They also show many of the basic approaches necessary to build generative models that the more advanced deep models will extend further. A linear factor model is deﬁned by the use of a stochastic, linear decoder function that generates x by adding noise to a linear transformation of h. These models are interesting because they allow us to discover explanatory factors that have a simple joint distribution. The simplicity of using a linear decoder made these models some of the ﬁrst latent variable models to be extensively studied. A linear factor model describes the data generation process as follows. First, we sample the explanatory factors h from a distribution h ∼ p(h), where p(h) is a factorial distribution, with p(h) = 491

(13.1)

i p(hi),

so that it is easy to

CHAPTER 13. LINEAR FACTOR MODELS

sample from. Next we sample the real-valued observable variables given the factors: x = W h + b + noise

(13.2)

where the noise is typically Gaussian and diagonal (independent across dimensions). This is illustrated in Fig. 13.1. h1

h2

h3

x1

x2

x3

x = W h + b + noise

Figure 13.1: The directed graphical model describing the linear factor model family, in which we assume that an observed data vector x is obtained by a linear combination of independent latent factors h, plus some noise. Diﬀerent model