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#![allow(clippy::double_parens)]
/*!
* (Automatic) Differentiation helpers
*
* # Automatic Differentiation
*
* This module provides structs for performing Forward and Reverse Automatic Differentiation
*
* ## Automatic Differentiation is not [Numerical Differentiation](https://en.wikipedia.org/wiki/Numerical_differentiation)
*
* You were probably introduced to differentiation as numeric differentiation,
* ie if you have a function 3x<sup>2</sup> then you can estimate its gradient
* at some value x by computing 3x<sup>2</sup> and 3(x+h)<sup>2</sup> where h
* is a very small value. The tangent line these two points create gives you an approximation
* of the gradient when you calculate (f(x+h) - f(x)) / h. Unfortunately floating
* point numbers in computers have limited precision, so this method is only approximate
* and can result in floating point errors. 1 + 1 might equal 2 but as you go smaller
* 10<sup>-i</sup> + 10<sup>-i</sup> starts to loook rather like 10<sup>-i</sup> as i goes
* into double digits.
*
* ## Automatic Differentiation is not Symbolic Differentiation
*
* If you were taught calculus you have probably done plenty of symbolic differentiation
* by hand. A function 3x<sup>2</sup> can be symbolically differentiated into 6x by applying
* simple rules to manipulate the algebra. Unfortunately the rules aren't so simple for
* more complex expressions such as [exponents](https://www.wolframalpha.com/input/?i=d%28x%5Ee%5E2%29%2Fdx),
* [logs](https://www.wolframalpha.com/input/?i=d%28log%28log%28x%29%29%29%2Fdx) or
* [trigonometry](https://www.wolframalpha.com/input/?i=d%28sin%28cos%28x%29%29%29%2Fdx).
* Symbolic differentiation can give you expressions which are just as or more complicated
* than the original, and doing it by hand can be error prone. Symbolic Differentiation is
* also tricky to relate to algorithmic computations that use control structures.
*
* ## [Automatic Differentiation](https://en.wikipedia.org/wiki/Automatic_differentiation)
*
* Automatic Differentiation computes the derivative of a function without rewriting
* the function as symbolic differentiation does and without the precision issues of numerical
* differentiation by splitting the derivative into lots of tiny steps of basic operations
* like addition and multiplication. These are combined using the chain rule. The downside
* is more memory is used than in symbolic or numerical differentiation, as derivatives have
* to be tracked through the computational graph.
*
* # Forward Differentiation
*
* Forward Differentiation computes all the gradients in a computational graph with respect
* to an input. For example, if you have a function f(x, y) = 5x<sup>3</sup> - 4x<sup>2</sup> +
* 10x - y, then for some actual value of x and y you can compute f(x,y) and δf(x,y)/δx
* together in one forward pass using forward differentiation. You can also make another pass
* and compute f(x,y) and δf(x,y)/δy for some actual value of x and y. It is possible to avoid
* redundantly calculating f(x,y) multiple times, but I am waiting on const generics to implement
* this. Regardless, forward differentiation requires making at least N+1 passes of the
* function to compute the derivatives of the output with respect to N inputs - and the current
* implementation will make 2N. However, you do get the gradients for every output in a
* single pass. This is poorly suited to neural nets as they often have a single output(loss)
* to differentiate many many inputs with respect to.
*
* # Reverse Mode Differentiation
*
* Reverse Mode Differentiation computes all the gradients in a computational graph for
* the same output. For example, if you have a function f(x, y) = 5x<sup>3</sup> -
* 4x<sup>2</sup> + 10x - y, then for some actual value of x and y you can compute f(x,y)
* and store all the intermediate results. You can then run a backward pass on the output
* of f(x, y) and obtain δf(x,y)/δx and δf(x,y)/δy for the actual values of x and y in a
* single pass. The catch is that reverse mode must store as many intermediate values as
* there are steps in the function which can use much more memory than forward mode.
* Reverse mode also requires making N backward passes to get the gradients for N different
* outputs. This is well suited to neural nets because we often have a single output (loss)
* to differentiate many inputs with respect to. However, reverse mode will be slower than
* forward mode if the number of inputs is small or there are many outputs.
*
* # Usage
*
* Both `Trace` and `Record` for forward and reverse automatic differentiation respectively
* implement `Numeric` and can generally be treated as normal numbers just like `f32` and `f64`.
*
* `Trace` is literally implemented as a dual number, and is more or less a one to one
* substitution. `Record` requires dynamically building a computational graph of the values
* and dependencies of each operation performed on them. This means performing operations on
* records have side effects, they add entries onto a `WengertList`. However, when using
* `Record` the side effects are abstracted away, just create a `WengertList` before you
* start creating Records.
*
* Given some function from N inputs to M outputs you can pass it `Trace`s or `Record`s
* and retrieve the first derivative from the outputs for all combinations of N and M.
* If N >> M then you should use `Record` as reverse mode automatic differentiation is
* much cheaper. If N << M then you should use `Trace` as it will be much cheaper. If
* you have large N and M, or small N and M, you might have to benchmark to find which
* method works best. However, most problems are N > M.
*
* For this example we use a function which takes two inputs, r and a, and returns two
* outputs, x and y.
*
* ## Using Trace
*
* ```
* use easy_ml::differentiation::Trace;
* use crate::easy_ml::numeric::extra::Cos;
* use crate::easy_ml::numeric::extra::Sin;
* fn cartesian(r: Trace<f32>, angle: Trace<f32>) -> (Trace<f32>, Trace<f32>) {
* let x = r * angle.cos();
* let y = r * angle.sin();
* (x, y)
* }
* // first find dx/dr and dy/dr
* let (x, y) = cartesian(Trace::variable(1.0), Trace::constant(2.0));
* let dx_dr = x.derivative;
* let dy_dr = y.derivative;
* // now find dx/da and dy/da
* let (x, y) = cartesian(Trace::constant(1.0), Trace::variable(2.0));
* let dx_da = x.derivative;
* let dy_da = y.derivative;
* ```
*
* ## Using Record
*
* ```
* use easy_ml::differentiation::{Record, WengertList};
* use crate::easy_ml::numeric::extra::{Cos, Sin};
* // the lifetimes tell the rust compiler that our inputs and outputs
* // can all live as long as the WengertList
* fn cartesian<'a>(r: Record<'a, f32>, angle: Record<'a, f32>)
* -> (Record<'a, f32>, Record<'a, f32>) {
* let x = r * angle.cos();
* let y = r * angle.sin();
* (x, y)
* }
* // first we must construct a WengertList to create records from
* let list = WengertList::new();
* let r = Record::variable(1.0, &list);
* let a = Record::variable(2.0, &list);
* let (x, y) = cartesian(r, a);
* // first find dx/dr and dx/da
* let x_derivatives = x.derivatives();
* let dx_dr = x_derivatives[&r];
* let dx_da = x_derivatives[&a];
* // now find dy/dr and dy/da
* let y_derivatives = y.derivatives();
* let dy_dr = y_derivatives[&r];
* let dy_da = y_derivatives[&a];
* ```
*
* ## Differences
*
* Notice how in the above examples all the same 4 derivatives are found, but in
* forward mode we rerun the function with a different input as the sole variable,
* the rest as constants, whereas in reverse mode we rerun the `derivatives()` function
* on a different output variable. With Reverse mode we would only pass constants into
* the `cartesian` function if we didn't want to get their derivatives (and avoid wasting
* memory on something we didn't need).
*
* ## Substitution
*
* There is no need to rewrite the input functions, as you can use the `Numeric` and `Real`
* traits to write a function that will take floating point numbers, `Trace`s and `Record`s.
*
* ```
* use easy_ml::differentiation::{Trace, Record, WengertList};
* use crate::easy_ml::numeric::Numeric;
* use crate::easy_ml::numeric::extra::{Real};
* fn cartesian<T: Numeric + Real + Copy>(r: T, angle: T) -> (T, T) {
* let x = r * angle.cos();
* let y = r * angle.sin();
* (x, y)
* }
* let list = WengertList::new();
* let r_record = Record::variable(1.0, &list);
* let a_record = Record::variable(2.0, &list);
* let (x_record, y_record) = cartesian(r_record, a_record);
* // find dx/dr using reverse mode automatic differentiation
* let x_derivatives = x_record.derivatives();
* let dx_dr_reverse = x_derivatives[&r_record];
* let (x_trace, y_trace) = cartesian(Trace::variable(1.0), Trace::constant(2.0));
* // now find dx/dr with forward automatic differentiation
* let dx_dr_forward = x_trace.derivative;
* assert_eq!(dx_dr_reverse, dx_dr_forward);
* let (x, y) = cartesian(1.0, 2.0);
* assert_eq!(x, x_record.number); assert_eq!(x, x_trace.number);
* assert_eq!(y, y_record.number); assert_eq!(y, y_trace.number);
* ```
*
* ## Equivalance
*
* Although in this example the derivatives found are identical, in practise, because
* forward and reverse mode compute things differently and floating point numbers have
* limited precision, you should not expect the derivatives to be exactly equal.
*
* # Further information
*
* - [Automatic Differentiation Step by Step](https://medium.com/@marksaroufim/automatic-differentiation-step-by-step-24240f97a6e6)
* - [Forward Mode Automatic Differentiation](https://en.wikipedia.org/wiki/Automatic_differentiation#Automatic_differentiation_using_dual_numbers)
* - [Reverse Mode Automatic Differentiation](https://rufflewind.com/2016-12-30/reverse-mode-automatic-differentiation)
* - [Automatic Differentiation: The most criminally underused tool in the potential machine learning toolbox?](https://justindomke.wordpress.com/2009/02/17/automatic-differentiation-the-most-criminally-underused-tool-in-the-potential-machine-learning-toolbox/)
* - [Yes you should understand backprop](https://medium.com/@karpathy/yes-you-should-understand-backprop-e2f06eab496b)
*/
pub mod operations;
pub mod record_operations;
pub mod trace_operations;
use crate::numeric::{Numeric, NumericRef};
#[cfg(feature = "serde")]
use serde::{Deserialize, Serialize};
/**
* A trait with no methods which is implemented for all primitive types.
*
* Importantly this trait is not implemented for Traces (or Records), to stop the compiler
* from trying to evaluate nested Traces of Traces or Records of Records as Numeric types.
* There is no reason to create a Trace of a Trace or Record of a Record, it won't do
* anything a Trace or Record can't except use more memory.
*
* The boilerplate implementations for primitives is performed with a macro.
* If a primitive type is missing from this list, please open an issue to add it in.
*/
pub trait Primitive {}
/**
* A dual number which traces a real number and keeps track of its derivative.
* This is used to perform Forward Automatic Differentiation
*
* Trace implements only first order differentiation. For example, given a function
* 3x<sup>2</sup>, you can use calculus to work out that its derivative with respect
* to x is 6x. You can also take the derivative of 6x with respect to x and work out
* that the second derivative is 6. By instead writing the function 3x<sup>2</sup> in
* code using Trace types as your numbers you can compute the first order derivative
* for a given value of x by passing your function `Trace { number: x, derivative: 1.0 }`.
*
* ```
* use easy_ml::differentiation::Trace;
* let x = Trace { number: 3.2, derivative: 1.0 };
* let dx = Trace::constant(3.0) * x * x;
* assert_eq!(dx.derivative, 3.2 * 6.0);
* ```
*
* Why the one for the starting derivative? Because δx/δx = 1, as with symbolic
* differentiation.
*
* # Acknowledgments
*
* The wikipedia page on [Automatic Differentiation](https://en.wikipedia.org/wiki/Automatic_differentiation)
* provided a very useful overview and explanation for understanding Forward Mode Automatic
* Differentiation as well as the implementation rules.
*/
#[derive(Debug)]
#[cfg_attr(feature = "serde", derive(Serialize, Deserialize))]
pub struct Trace<T: Primitive> {
/**
* The real number
*/
pub number: T,
/**
* The first order derivative of this number.
*/
pub derivative: T,
}
/**
* The main set of methods for using Trace types for Forward Differentiation.
*
* The general steps are
* 1. create one variable
* 2. create as many constants as needed
* 3. do operations on the variable and constants
* 4. the outputs will have derivatives computed which can be accessed from
* the `.derivative` field, with each derivative being the output with respect
* to the input variable.
* 5. if you need derivatives for a different input then do everything all over again
* or do them all in parallel
*/
impl<T: Numeric + Primitive> Trace<T> {
/**
* Constants are lifted to Traces with a derivative of 0
*
* Why zero for the starting derivative? Because for any constant C
* δC/δx = 0, as with symbolic differentiation.
*/
pub fn constant(c: T) -> Trace<T> {
Trace {
number: c,
derivative: T::zero(),
}
}
/**
* To lift a variable that you want to find the derivative of
* a function to, the Trace starts with a derivative of 1
*
* Why the one for the starting derivative? Because δx/δx = 1, as with symbolic
* differentiation.
*/
pub fn variable(x: T) -> Trace<T> {
Trace {
number: x,
derivative: T::one(),
}
}
/**
* Computes the derivative of a function with respect to its input x.
*
* This is a shorthand for `(function(Trace::variable(x))).derivative`
*
* In the more general case, if you provide a function with an input x
* and it returns N outputs y<sub>1</sub> to y<sub>N</sub> then you
* have computed all the derivatives δy<sub>i</sub>/δx for i = 1 to N.
*/
pub fn derivative(function: impl FnOnce(Trace<T>) -> Trace<T>, x: T) -> T {
(function(Trace::variable(x))).derivative
}
}
impl<T: Numeric + Primitive> Trace<T>
where
for<'a> &'a T: NumericRef<T>,
{
/**
* Creates a new Trace from a reference to an existing Trace by applying
* some unary function to it which operates on the type the Trace wraps.
*
* To compute the new trace, the unary function of some input x to some
* output y is needed along with its derivative with respect to its input x.
*
* For example, tanh is a commonly used activation function, but the Real trait
* does not include this operation and Trace has no operations for it specifically.
* However, you can use this function to compute the tanh of a Trace like so:
*
* ```
* use easy_ml::differentiation::Trace;
* let x = Trace::variable(0.7f32);
* // the derivative of tanh(x) is sech(x) * sech(x) which is equivalent to
* // 1 / (cosh(x) * cosh(x))
* let y = x.unary(|x| x.tanh(), |x| 1.0 / (x.cosh() * x.cosh()));
* assert_eq!(y.derivative, 1.0f32 / (0.7f32.cosh() * 0.7f32.cosh()));
* ```
*/
#[inline]
pub fn unary(&self, fx: impl Fn(T) -> T, dfx_dx: impl Fn(T) -> T) -> Trace<T> {
Trace {
number: fx(self.number.clone()),
derivative: self.derivative.clone() * dfx_dx(self.number.clone()),
}
}
/**
* Creates a new Trace from a reference to two existing Traces by applying
* some binary function to them which operates on two arguments of the type
* the Traces wrap.
*
* To compute the new trace, the binary function of some inputs x and y to some
* output z is needed along with its derivative with respect to its first input x and
* its derivative with respect to its second input y.
*
* For example, atan2 takes two arguments, but the Real trait
* does not include this operation and Trace has no operations for it specifically.
* However, you can use this function to compute the atan2 of two Traces like so:
*
* ```
* use easy_ml::differentiation::Trace;
* let x = Trace::variable(3.0f32);
* let y = Trace::variable(3.0f32);
* // the derivative of atan2 with respect to x is y/(x*x + y*y)
* // https://www.wolframalpha.com/input/?i=d%28atan2%28x%2Cy%29%29%2Fdx
* // the derivative of atan2 with respect to y is -x/(x*x + y*y)
* // https://www.wolframalpha.com/input/?i=d%28atan2%28x%2Cy%29%29%2Fdy
* let z = x.binary(&y,
* |x, y| x.atan2(y),
* |x, y| y/((x*x) + (y*y)),
* |x, y| -x/((x*x) + (y*y))
* );
* ```
*/
#[inline]
pub fn binary(
&self,
rhs: &Trace<T>,
fxy: impl Fn(T, T) -> T,
dfxy_dx: impl Fn(T, T) -> T,
dfxy_dy: impl Fn(T, T) -> T,
) -> Trace<T> {
Trace {
number: fxy(self.number.clone(), rhs.number.clone()),
#[rustfmt::skip]
derivative: (
((self.derivative.clone() * dfxy_dx(self.number.clone(), rhs.number.clone()))
+ (rhs.derivative.clone() * dfxy_dy(self.number.clone(), rhs.number.clone())))
),
}
}
}
use std::cell::RefCell;
type Index = usize;
/**
* A list of operations performed in a forward pass of a dynamic computational graph,
* used for Reverse Mode Automatic Differentiation.
*
* This is dynamic, as in, you build the [Wengert list](https://en.wikipedia.org/wiki/Automatic_differentiation#Reverse_accumulation)
* at runtime by performing operations like addition and multiplication on
* [Records](Record) that were created with that Wengert list.
*
* When you perform a backward pass to obtain the gradients you travel back up the
* computational graph using the stored intermediate values from this list to compute
* all the gradients of the inputs and every intermediate step with respect to an output.
*
* Although sophisticated implementations can make the Wengert list only log(N) in length
* by storing only some of the intermediate steps of N computational steps, this implementation
* is not as sophisticated, and will store all of them.
*
* # Panics
*
* Every operation and nearly every method a Record has involves manipulating the
* record's history on its referenced WengertList. This WengertList itself maintains
* a [RefCell](std::cell::RefCell) which tracks
* borrows at runtime rather than compile time. This is neccessary to maintain the
* illusion that Records are just ordinary numbers, and the side effects of doing
* arithmetic with Records are limited to their referenced WengertList. Hence, the Rust
* compiler infers that it is not safe to share references to WengertLists between threads,
* nor transfer Records across threads. If you called a method on two Records that both
* mutably borrowed from the same WengertList at once, which could be trivially done with
* multiple threads, then the code would panic. Easy ML shouldn't allow you to do this
* in safe Rust because each mutable borrow of the WengertList is dropped at the end of each
* Record method call, and you can't call two methods simulatenously without threading.
*/
#[derive(Debug)]
pub struct WengertList<T> {
// It is neccessary to wrap the vec in a RefCell to allow for mutating
// this list from immutable references held by each
operations: RefCell<Vec<Operation<T>>>,
}
/**
* A binary operation to record on a WengertList. For unary operations the
* right derivative is set to 0, and for nullary operations both derivatives
* are set to 0.
*
* Each operation acts like a node in an upside down binary tree, with two parents that
* each node was computed from. The main difference is that the numerical
* index of those parents in the WengertList is stored, rather than any pointers.
*/
#[derive(Debug)]
struct Operation<T> {
left_parent: Index,
right_parent: Index,
left_derivative: T,
right_derivative: T,
}
/**
* Computed derivatives of a computational graph for some output [Record] variable.
*
* This can be indexed by any Record used in the computational graph to get
* the derivative with respect to that input.
*
* Indexing using Records not involved in the computational graph, or involved
* in a different one will return nonsense or index out of bounds and panic. In
* the future this may be changed to always panic.
*/
#[derive(Debug)]
pub struct Derivatives<T> {
derivatives: Vec<T>,
}
/**
* Any derivatives of a Cloneable type implements clone
*/
impl<T: Clone> Clone for Derivatives<T> {
fn clone(&self) -> Self {
Derivatives {
derivatives: self.derivatives.clone(),
}
}
}
impl<T: Clone + Primitive> Derivatives<T> {
/**
* Quries the derivative at the provided record as input.
*
* If you construct a Derivatives object for some output y,
* and call .at(&x) on it for some input x, this returns dy/dx.
*/
pub fn at(&self, input: &Record<T>) -> T {
self.derivatives[input.index].clone()
}
}
impl<'a, T: Primitive> std::ops::Index<&Record<'a, T>> for Derivatives<T> {
type Output = T;
/**
* Quries the derivative at the provided record as input.
*
* If you construct a Derivatives object for some output y,
* and call .at(&x) on it for some input x, this returns dy/dx.
*/
fn index(&self, input: &Record<'a, T>) -> &Self::Output {
&self.derivatives[input.index]
}
}
impl<T> std::convert::From<Derivatives<T>> for Vec<T> {
/**
* Converts the Derivatives struct into a Vec of derivatives that
* can be indexed with `usize`s. The indexes correspond to the
* index field on Records.
*/
fn from(derivatives: Derivatives<T>) -> Self {
derivatives.derivatives
}
}
/**
* Any operation of a Cloneable type implements clone
*/
impl<T: Clone + Primitive> Clone for Operation<T> {
fn clone(&self) -> Self {
Operation {
left_parent: self.left_parent,
right_parent: self.right_parent,
left_derivative: self.left_derivative.clone(),
right_derivative: self.right_derivative.clone(),
}
}
}
/**
* A wrapper around a real number which records it going through the computational
* graph. This is used to perform Reverse Mode Automatic Differentiation.
*
* # Panics
*
* Every operation and nearly every method a Record has involves manipulating the
* record's history on its referenced [WengertList]. This WengertList itself maintains
* a [RefCell](std::cell::RefCell) which tracks
* borrows at runtime rather than compile time. This is neccessary to maintain the
* illusion that Records are just ordinary numbers, and the side effects of doing
* arithmetic with Records are limited to their referenced WengertList. Hence, the Rust
* compiler infers that it is not safe to share references to WengertLists between threads,
* nor transfer Records across threads. If you called a method on two Records that both
* mutably borrowed from the same WengertList at once, which could be trivially done with
* multiple threads, then the code would panic. Easy ML shouldn't allow you to do this
* in safe Rust because each mutable borrow of the WengertList is dropped at the end of each
* Record method call, and you can't call two methods simulatenously without threading.
*
* # Acknowledgments
*
* A [tutorial by Rufflewind](https://rufflewind.com/2016-12-30/reverse-mode-automatic-differentiation)
* and the associated [MIT licensed](http://opensource.org/licenses/MIT)
* [soure code](https://github.com/Rufflewind/revad/blob/master/src/tape.rs) were invaluable
* in providing understanding on how to implement Reverse Mode Automatic Differentiation.
*/
#[derive(Debug)]
pub struct Record<'a, T: Primitive> {
// A record consists of a number used in the forward pass, as
// well as a WengertList of operations performed on the numbers
// and each record needs to know which point in the history they
// are for.
/**
* The real number
*/
pub number: T,
history: Option<&'a WengertList<T>>,
/**
* The index of this number in its [WengertList]. The first entry will be 0,
* the next 1 and so on.
*
* In normal use cases you should not need to read this field,
* you can index [Derivatives] directly with Records.
*/
pub index: Index,
}
/**
* The main set of methods for using Record types for Reverse Differentiation.
*
* The general steps are
* 1. create a `WengertList`
* 2. create variables from this list
* 3. do operations on the variables
* 4. from the output you want to compute derivatives for call `.derivatives()`
* 5. index the `Derivatives` object with the index variables to get the derivatives
* with respect to each input
* 6. if you want to make another pass call `clear()` on the `WengertList`
* and then call `reset()` on all of the variables to forget the gradients already
* computed (the order of `clear` then `reset` is very important!).
*
* Constants can be used to save memory if you have numbers that
* you do not need to compute the gradients with respect to.
*/
impl<'a, T: Numeric + Primitive> Record<'a, T> {
/**
* Creates an untracked Record which has no backing WengertList.
*
* This is provided for using constants along with Records in operations.
*
* For example with y = x + 4 the computation graph could be conceived as
* a y node with parent nodes of x and 4 combined with the operation +.
* However there is no need to record the derivatives of a constant, so
* instead the computation graph can be conceived as a y node with a single
* parent node of x and the unary operation of +4.
*
* This is also used for the type level constructors required by Numeric
* which are also considered constants.
*/
pub fn constant(c: T) -> Record<'a, T> {
Record {
number: c,
history: None,
index: 0,
}
}
/**
* Creates a record backed by the provided WengertList.
*
* The record cannot live longer than the WengertList, hence
* the following example does not compile
*
* ```compile_fail
* use easy_ml::differentiation::Record;
* use easy_ml::differentiation::WengertList;
* let record = {
* let list = WengertList::new();
* Record::variable(1.0, &list)
* }; // list no longer in scope
* ```
*
* You can alternatively use the [record constructor on the WengertList type](WengertList::variable()).
*/
pub fn variable(x: T, history: &'a WengertList<T>) -> Record<'a, T> {
Record {
number: x,
history: Some(history),
index: history.append_nullary(),
}
}
/**
* Resets this Record to place it back on its WengertList, for use
* in performing another derivation after clearing the WengertList.
*/
pub fn reset(&mut self) {
match self.history {
None => (), // noop
Some(history) => self.index = history.append_nullary(),
};
}
/**
* A convenience helper function which takes a Record by value and
* calls [reset](Record::reset()) on it.
*/
pub fn do_reset(mut x: Record<T>) -> Record<T> {
x.reset();
x
}
}
impl<'a, T: Numeric + Primitive> Record<'a, T>
where
for<'t> &'t T: NumericRef<T>,
{
/**
* Performs a backward pass up this record's WengertList from this
* record as the output, computing all the derivatives for the inputs
* involving this output.
*
* If you have N inputs x<sub>1</sub> to x<sub>N</sub>, and this output is y,
* then this computes all the derivatives δy/δx<sub>i</sub> for i = 1 to N.
*
* # Panics
*
* Panics if the Record has no backing WengertList, ie it was created as a
* constant.
*/
#[track_caller]
pub fn derivatives(&self) -> Derivatives<T> {
let history = match self.history {
None => panic!("Record has no WengertList to find derivatives from"),
Some(h) => h,
};
let operations = history.operations.borrow();
let mut derivatives = vec![T::zero(); operations.len()];
// δy/δy = 1
derivatives[self.index] = T::one();
// Go back up the computation graph to the inputs
for i in (0..operations.len()).rev() {
let operation = operations[i].clone();
let derivative = derivatives[i].clone();
// The chain rule allows breaking up the derivative of the output y
// with respect to the input x into many smaller derivatives that
// are summed together.
// δy/δx = δy/δw * δw/δx
// δy/δx = sum for all i parents of y ( δy/δw_i * δw_i/δx )
derivatives[operation.left_parent] = derivatives[operation.left_parent].clone()
+ derivative.clone() * operation.left_derivative;
derivatives[operation.right_parent] = derivatives[operation.right_parent].clone()
+ derivative * operation.right_derivative;
}
Derivatives { derivatives }
}
}
impl<T: Primitive> WengertList<T> {
/**
* Creates a new empty WengertList from which Records can be constructed.
*/
pub fn new() -> WengertList<T> {
WengertList {
operations: RefCell::new(Vec::new()),
}
}
}
impl<T: Primitive> Default for WengertList<T> {
fn default() -> Self {
Self::new()
}
}
impl<T> WengertList<T> {
/**
* Clears a WengertList to make it empty again. After clearing a WengertList
* you must reset all the Records still using that list. Then you can perform
* another computation and get new gradients.
*/
pub fn clear(&self) {
self.operations.borrow_mut().clear();
}
}
impl<T: Numeric + Primitive> WengertList<T> {
/**
* Creates a record backed by this WengertList.
*
* You can alternatively use the [record constructor on the Record type](Record::variable()).
*/
pub fn variable(&self, x: T) -> Record<T> {
Record {
number: x,
history: Some(self),
index: self.append_nullary(),
}
}
/**
* Adds a value to the list which does not have any parent values.
*/
fn append_nullary(&self) -> Index {
let mut operations = self.operations.borrow_mut();
// insert into end of list
let index = operations.len();
operations.push(Operation {
// this index of the child is used for both indexes as these
// won't be needed but will always be valid (ie point to a
// real entry on the list)
left_parent: index,
right_parent: index,
// for the parents 0 is used to zero out these calculations
// as there are no parents
left_derivative: T::zero(),
right_derivative: T::zero(),
});
index
}
/**
* Adds a value to the list which has one parent.
*
* For an output w_N which depends on one parent w_N-1
* the derivative cached here is δw_N / δw_N-1
*
* For example, if z = sin(x), then δz/δx = cos(x)
*/
fn append_unary(&self, parent: Index, derivative: T) -> Index {
let mut operations = self.operations.borrow_mut();
// insert into end of list
let index = operations.len();
operations.push(Operation {
left_parent: parent,
// this index of the child is used as this index won't be needed
// but will always be valid (ie points to a real entry on the list)
right_parent: index,
left_derivative: derivative,
// for the right parent 0 is used to zero out this calculation
// as there is no right parent
right_derivative: T::zero(),
});
index
}
/**
* Adds a value to the list which has two parents.
*
* For an output w_N which depends on two parents w_N-1
* and w_N-2 the derivatives cached here are δw_N / δw_N-1
* and δw_N / δw_N-2.
*
* For example, if z = y + x, then δz/δy = 1 and δz/δx = 1
* For example, if z = y * x, then δz/δy = x and δz/δ/x = y
*/
fn append_binary(
&self,
left_parent: Index,
left_derivative: T,
right_parent: Index,
right_derivative: T,
) -> Index {
let mut operations = self.operations.borrow_mut();
// insert into end of list
let index = operations.len();
operations.push(Operation {
left_parent,
right_parent,
left_derivative,
right_derivative,
});
index
}
}
/**
* Any Wengert list of a Cloneable type implements clone
*/
impl<T: Clone + Primitive> Clone for WengertList<T> {
fn clone(&self) -> Self {
WengertList {
operations: RefCell::new(self.operations.borrow().clone()),
}
}
}
impl<'a, T: Numeric + Primitive> Record<'a, T>
where
for<'t> &'t T: NumericRef<T>,
{
/**
* Creates a new Record from a reference to an existing Record by applying
* some unary function to it which operates on the type the Record wraps.
*
* To compute the new record, the unary function of some input x to some
* output y is needed along with its derivative with respect to its input x.
*
* For example, tanh is a commonly used activation function, but the Real trait
* does not include this operation and Record has no operations for it specifically.
* However, you can use this function to compute the tanh of a Record like so:
*
* ```
* use easy_ml::differentiation::{Record, WengertList};
* let list = WengertList::new();
* let x = Record::variable(0.7f32, &list);
* // the derivative of tanh(x) is sech(x) * sech(x) which is equivalent to
* // 1 / (cosh(x) * cosh(x))
* let y = x.unary(|x| x.tanh(), |x| 1.0 / (x.cosh() * x.cosh()));
* assert_eq!(y.derivatives()[&x], 1.0f32 / (0.7f32.cosh() * 0.7f32.cosh()));
* ```
*/
#[inline]
pub fn unary(&self, fx: impl Fn(T) -> T, dfx_dx: impl Fn(T) -> T) -> Record<T> {
match self.history {
None => Record {
number: fx(self.number.clone()),
history: None,
index: 0,
},
Some(history) => Record {
number: fx(self.number.clone()),
history: Some(history),
index: history.append_unary(self.index, dfx_dx(self.number.clone())),
},
}
}
/**
* Creates a new Record from a reference to two existing Records by applying
* some binary function to them which operates on two arguments of the type
* the Records wrap.
*
* To compute the new record, the binary function of some inputs x and y to some
* output z is needed along with its derivative with respect to its first input x and
* its derivative with respect to its second input y.
*
* For example, atan2 takes two arguments, but the Real trait
* does not include this operation and Record has no operations for it specifically.
* However, you can use this function to compute the atan2 of two Records like so:
*
* ```
* use easy_ml::differentiation::{Record, WengertList};
* let list = WengertList::new();
* let x = Record::variable(3.0f32, &list);
* let y = Record::variable(3.0f32, &list);
* // the derivative of atan2 with respect to x is y/(x*x + y*y)
* // https://www.wolframalpha.com/input/?i=d%28atan2%28x%2Cy%29%29%2Fdx
* // the derivative of atan2 with respect to y is -x/(x*x + y*y)
* // https://www.wolframalpha.com/input/?i=d%28atan2%28x%2Cy%29%29%2Fdy
* let z = x.binary(&y,
* |x, y| x.atan2(y),
* |x, y| y/((x*x) + (y*y)),
* |x, y| -x/((x*x) + (y*y))
* );
* let derivatives = z.derivatives();
* let dx = derivatives[&x];
* let dy = derivatives[&y];
* ```
*/
#[inline]
#[track_caller]
pub fn binary(
&self,
rhs: &Record<'a, T>,
fxy: impl Fn(T, T) -> T,
dfxy_dx: impl Fn(T, T) -> T,
dfxy_dy: impl Fn(T, T) -> T,
) -> Record<T> {
assert!(
record_operations::same_list(self, rhs),
"Records must be using the same WengertList"
);
match (self.history, rhs.history) {
(None, None) => Record {
number: fxy(self.number.clone(), rhs.number.clone()),
history: None,
index: 0,
},
(Some(history), None) => Record {
number: fxy(self.number.clone(), rhs.number.clone()),
history: Some(history),
index: history.append_unary(
// if rhs didn't have a history, don't track that derivative
self.index,
dfxy_dx(self.number.clone(), rhs.number.clone()),
),
},
(None, Some(history)) => Record {
number: fxy(self.number.clone(), rhs.number.clone()),
history: Some(history),
index: history.append_unary(
// if self didn't have a history, don't track that derivative
rhs.index,
dfxy_dy(self.number.clone(), rhs.number.clone()),
),
},
(Some(history), Some(_)) => Record {
number: fxy(self.number.clone(), rhs.number.clone()),
history: Some(history),
index: history.append_binary(
self.index,
dfxy_dx(self.number.clone(), rhs.number.clone()),
rhs.index,
dfxy_dy(self.number.clone(), rhs.number.clone()),
),
},
}
}
}
#[cfg(test)]
#[should_panic]
#[test]
fn test_record_derivatives_when_no_history() {
let record = Record::constant(1.0);
record.derivatives();
}
#[test]
fn test_sync() {
fn assert_sync<T: Sync>() {}
assert_sync::<Trace<f64>>();
}
#[test]
fn test_send() {
fn assert_send<T: Send>() {}
assert_send::<Trace<f64>>();
}