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//! Parameter optimization.
//!
//! The fit of the parameters is done by gradient descent (using the ADAM algorithm) on the gradient of the marginal log-likelihood
//! (which let us use all the data without bothering with cross-validation).
//!
//! If the kernel can be rescaled, we use ideas from [Fast methods for training Gaussian processes on large datasets](https://arxiv.org/pdf/1604.01250.pdf)
//! to rescale the kernel at each step with the optimal magnitude which has the effect of fitting the noise without computing its gradient.
//!
//! Otherwise we fit the noise in log-scale as its magnitude matters more than its precise value.
use chrono::{Duration, Utc};
use super::GaussianProcess;
use crate::algebra::{make_cholesky_cov_matrix, make_gradient_covariance_matrices};
use crate::parameters::{kernel::Kernel, prior::Prior};
impl<KernelType: Kernel, PriorType: Prior> GaussianProcess<KernelType, PriorType>
{
//-------------------------------------------------------------------------------------------------
// NON-SCALABLE KERNEL
/// Computes the gradient of the marginal likelihood for the current value of each parameter.
/// The produced vector contains the gradient per kernel parameter followed by the gradient for the noise parameter.
fn gradient_marginal_likelihood(&self) -> Vec<f64>
{
// formula: 1/2 ( transpose(alpha) * dp * alpha - trace(K^-1 * dp) )
// K = cov(train,train)
// alpha = K^-1 * output
// dp = gradient(K, parameter)
// Needed for the per parameter gradient computation.
let cov_inv = self.covmat_cholesky.inverse();
let alpha = &cov_inv * self.training_outputs.as_vector();
// Loop over the gradient matrix for each parameter.
let mut results = vec![];
for cov_gradient in make_gradient_covariance_matrices(&self.training_inputs.as_matrix(), &self.kernel)
{
// transpose(alpha) * cov_gradient * alpha
let data_fit: f64 = cov_gradient.column_iter()
.zip(alpha.iter())
.map(|(col, alpha_col)| alpha.dot(&col) * alpha_col)
.sum();
// trace(cov_inv * cov_gradient)
let complexity_penalty: f64 =
cov_inv.row_iter().zip(cov_gradient.column_iter()).map(|(c, d)| c.tr_dot(&d)).sum();
results.push((data_fit - complexity_penalty) / 2.);
}
// Adds the noise parameter.
// gradient(K, noise) = 2*noise*Id
let data_fit = alpha.dot(&alpha);
let complexity_penalty = cov_inv.trace();
let noise_gradient = self.noise * (data_fit - complexity_penalty);
results.push(noise_gradient);
results
}
/// Fit parameters using a gradient descent algorithm.
///
/// Runs for a maximum of `max_iter` iterations (100 is a good default value).
/// Stops prematurely if all the components of the gradient go below `convergence_fraction` time the value of their respectively parameter (0.05 is a good default value).
/// Stops prematurely if the runtime exceeds `max_time`.
///
/// The `noise` parameter is fitted in log-scale as its magnitude matters more than its precise value.
pub(super) fn optimize_parameters(&mut self,
max_iter: usize,
convergence_fraction: f64,
max_time: Duration)
{
// use the ADAM gradient descent algorithm
// see [optimizing-gradient-descent](https://ruder.io/optimizing-gradient-descent/)
// for a good point on current gradient descent algorithms
// Constant parameters.
let beta1 = 0.9;
let beta2 = 0.999;
let epsilon = 1e-8;
let learning_rate = 0.1;
let mut parameters: Vec<_> = self.kernel
.get_parameters()
.iter()
.map(|&p| {
if p == 0.
{
epsilon
}
else
{
p
}
}) // Insures no parameter is 0 (which would block the algorithm).
.collect();
parameters.push(self.noise.ln()); // Adds noise in log-space.
let mut mean_grad = vec![0.; parameters.len()];
let mut var_grad = vec![0.; parameters.len()];
let time_start = Utc::now();
for i in 1..=max_iter
{
let mut gradients = self.gradient_marginal_likelihood();
if let Some(noise_grad) = gradients.last_mut()
{
// Corrects gradient of noise for log-space.
*noise_grad *= self.noise
}
let mut had_significant_progress = false;
for p in 0..parameters.len()
{
mean_grad[p] = beta1 * mean_grad[p] + (1. - beta1) * gradients[p];
var_grad[p] = beta2 * var_grad[p] + (1. - beta2) * gradients[p].powi(2);
let bias_corrected_mean = mean_grad[p] / (1. - beta1.powi(i as i32));
let bias_corrected_variance = var_grad[p] / (1. - beta2.powi(i as i32));
let delta = learning_rate * bias_corrected_mean / (bias_corrected_variance.sqrt() + epsilon);
had_significant_progress |= delta.abs() > convergence_fraction;
parameters[p] *= 1. + delta;
}
// Sets parameters.
self.kernel.set_parameters(¶meters);
if let Some(noise) = parameters.last()
{
// Gets out of log-space before setting noise.
self.noise = noise.exp()
}
// Fits model.
self.covmat_cholesky = make_cholesky_cov_matrix(&self.training_inputs.as_matrix(),
&self.kernel,
self.noise,
self.cholesky_epsilon);
if (!had_significant_progress) || (Utc::now().signed_duration_since(time_start) > max_time)
{
//println!("Iterations:{}", i);
break;
};
}
/*println!("Fit done. likelihood:{} parameters:{:?} noise:{:e}",
self.likelihood(),
parameters,
self.noise);*/
}
//-------------------------------------------------------------------------------------------------
// SCALABLE KERNEL
/// Returns a couple containing the optimal scale for the kernel+noise (which is used to optimize the noise)
/// plus a vector containing the gradient per kernel parameter (but NOT the gradient for the noise parameter).
///
/// See [Fast methods for training Gaussian processes on large datasets](https://arxiv.org/pdf/1604.01250.pdf)
/// for the formula used to compute the scale and the modification to the gradient.
fn scaled_gradient_marginal_likelihood(&self) -> (f64, Vec<f64>)
{
// formula:
// gradient = 1/2 ( transpose(alpha) * dp * alpha / scale - trace(K^-1 * dp) )
// scale = transpose(output) * K^-1 * output / n
// K = cov(train,train)
// alpha = K^-1 * output
// dp = gradient(K, parameter)
// Needed for the per parameter gradient computation.
let cov_inv = self.covmat_cholesky.inverse();
let training_output = self.training_outputs.as_vector();
let alpha = &cov_inv * training_output;
// Scaling for the kernel.
let scale = training_output.dot(&alpha) / (training_output.nrows() as f64);
// Loop on the gradient matrix for each parameter.
let mut results = vec![];
for cov_gradient in make_gradient_covariance_matrices(&self.training_inputs.as_matrix(), &self.kernel)
{
// transpose(alpha) * cov_gradient * alpha / scale
// NOTE: This quantity is divided by the scale which is not the case for the unscaled gradient.
let data_fit = cov_gradient.column_iter()
.zip(alpha.iter())
.map(|(col, alpha_col)| alpha.dot(&col) * alpha_col)
.sum::<f64>()
/ scale;
// trace(cov_inv * cov_gradient)
let complexity_penalty: f64 =
cov_inv.row_iter().zip(cov_gradient.column_iter()).map(|(c, d)| c.tr_dot(&d)).sum();
results.push((data_fit - complexity_penalty) / 2.);
}
// adds the noise parameter
// gradient(K, noise) = 2*noise*Id
/*let data_fit = alpha.dot(&alpha) / scale;
let complexity_penalty = cov_inv.trace();
let noise_gradient = self.noise * (data_fit - complexity_penalty);
results.push(noise_gradient);*/
(scale, results)
}
/// Fit parameters using a gradient descent algorithm.
/// Additionally, at each step, the kernel and noise are rescaled using the optimal magnitude.
///
/// Runs for a maximum of `max_iter` iterations (100 is a good default value).
/// Stops prematurely if all the components of the gradient go below `convergence_fraction` time the value of their respectively parameter (0.05 is a good default value).
/// Stops prematurely if the runtime exceeds `max_time`.
pub(super) fn scaled_optimize_parameters(&mut self,
max_iter: usize,
convergence_fraction: f64,
max_time: Duration)
{
// use the ADAM gradient descent algorithm
// see [optimizing-gradient-descent](https://ruder.io/optimizing-gradient-descent/)
// for a good point on current gradient descent algorithms
// Constant parameters.
let beta1 = 0.9;
let beta2 = 0.999;
let epsilon = 1e-8;
let learning_rate = 0.1;
let mut parameters: Vec<_> = self.kernel
.get_parameters()
.iter()
.map(|&p| {
if p == 0.
{
epsilon
}
else
{
p
}
}) // Insures no parameter is 0 (which would block the algorithm).
.collect();
let mut mean_grad = vec![0.; parameters.len()];
let mut var_grad = vec![0.; parameters.len()];
let time_start = Utc::now();
for i in 1..=max_iter
{
let (scale, gradients) = self.scaled_gradient_marginal_likelihood();
let mut had_significant_progress = false;
for p in 0..parameters.len()
{
mean_grad[p] = beta1 * mean_grad[p] + (1. - beta1) * gradients[p];
var_grad[p] = beta2 * var_grad[p] + (1. - beta2) * gradients[p].powi(2);
let bias_corrected_mean = mean_grad[p] / (1. - beta1.powi(i as i32));
let bias_corrected_variance = var_grad[p] / (1. - beta2.powi(i as i32));
let delta = learning_rate * bias_corrected_mean / (bias_corrected_variance.sqrt() + epsilon);
had_significant_progress |= delta.abs() > convergence_fraction;
parameters[p] *= 1. + delta;
}
// Set parameters.
self.kernel.set_parameters(¶meters);
self.kernel.rescale(scale);
self.noise *= scale;
parameters = self.kernel.get_parameters(); // Get parameters back as they have been rescaled.
// Fits model.
self.covmat_cholesky = make_cholesky_cov_matrix(&self.training_inputs.as_matrix(),
&self.kernel,
self.noise,
self.cholesky_epsilon);
if (!had_significant_progress) || (Utc::now().signed_duration_since(time_start) > max_time)
{
//println!("Iterations:{}", i);
break;
};
}
/*println!("Scaled fit done. likelihood:{} parameters:{:?} noise:{:e}",
self.likelihood(),
parameters,
self.noise);*/
}
}