Source code for gpytorch.mlls.leave_one_out_pseudo_likelihood

#!/usr/bin/env python3
import math

import torch
from torch import Tensor

from ..distributions import MultivariateNormal
from .exact_marginal_log_likelihood import ExactMarginalLogLikelihood

[docs]class LeaveOneOutPseudoLikelihood(ExactMarginalLogLikelihood): r""" The leave one out cross-validation (LOO-CV) likelihood from RW 5.4.2 for an exact Gaussian process with a Gaussian likelihood. This offers an alternative to the exact marginal log likelihood where we instead maximize the sum of the leave one out log probabilities :math:`\log p(y_i | X, y_{-i}, \theta)`. Naively, this will be O(n^4) with Cholesky as we need to compute `n` Cholesky factorizations. Fortunately, given the Cholesky factorization of the full kernel matrix (without any points removed), we can compute both the mean and variance of each removed point via a bordered system formulation making the total complexity O(n^3). The LOO-CV approach can be more robust against model mis-specification as it gives an estimate for the (log) predictive probability, whether or not the assumptions of the model is fulfilled. .. note:: This module will not work with anything other than a :obj:`~gpytorch.likelihoods.GaussianLikelihood` and a :obj:`~gpytorch.models.ExactGP`. It also cannot be used in conjunction with stochastic optimization. :param ~gpytorch.likelihoods.GaussianLikelihood likelihood: The Gaussian likelihood for the model :param ~gpytorch.models.ExactGP model: The exact GP model Example: >>> # model is a gpytorch.models.ExactGP >>> # likelihood is a gpytorch.likelihoods.Likelihood >>> loocv = gpytorch.mlls.LeaveOneOutPseudoLikelihood(likelihood, model) >>> >>> output = model(train_x) >>> loss = -loocv(output, train_y) >>> loss.backward() """ def __init__(self, likelihood, model): super().__init__(likelihood=likelihood, model=model) self.likelihood = likelihood self.model = model
[docs] def forward(self, function_dist: MultivariateNormal, target: Tensor, *params) -> Tensor: r""" Computes the leave one out likelihood given :math:`p(\mathbf f)` and `\mathbf y` :param ~gpytorch.distributions.MultivariateNormal output: the outputs of the latent function (the :obj:`~gpytorch.models.GP`) :param torch.Tensor target: :math:`\mathbf y` The target values :param dict kwargs: Additional arguments to pass to the likelihood's forward function. """ output = self.likelihood(function_dist, *params) m, L = output.mean, output.lazy_covariance_matrix.cholesky(upper=False) m = m.reshape(*target.shape) identity = torch.eye(*L.shape[-2:], dtype=m.dtype, device=m.device) sigma2 = 1.0 / L._cholesky_solve(identity, upper=False).diagonal(dim1=-1, dim2=-2) # 1 / diag(inv(K)) mu = target - L._cholesky_solve((target - m).unsqueeze(-1), upper=False).squeeze(-1) * sigma2 term1 = -0.5 * sigma2.log() term2 = -0.5 * (target - mu).pow(2.0) / sigma2 res = (term1 + term2).sum(dim=-1) res = self._add_other_terms(res, params) # Scale by the amount of data we have and then add on the scaled constant num_data = target.size(-1) return res.div_(num_data) - 0.5 * math.log(2 * math.pi)