Source code for gpytorch.mlls.variational_elbo

#!/usr/bin/env python3

from ._approximate_mll import _ApproximateMarginalLogLikelihood

[docs]class VariationalELBO(_ApproximateMarginalLogLikelihood): r""" The variational evidence lower bound (ELBO). This is used to optimize variational Gaussian processes (with or without stochastic optimization). .. math:: \begin{align*} \mathcal{L}_\text{ELBO} &= \mathbb{E}_{p_\text{data}( y, \mathbf x )} \left[ \mathbb{E}_{p(f \mid \mathbf u, \mathbf x) q(\mathbf u)} \left[ \log p( y \! \mid \! f) \right] \right] - \beta \: \text{KL} \left[ q( \mathbf u) \Vert p( \mathbf u) \right] \\ &\approx \sum_{i=1}^N \mathbb{E}_{q( f_i)} \left[ \log p( y_i \! \mid \! f_i) \right] - \beta \: \text{KL} \left[ q( \mathbf u) \Vert p( \mathbf u) \right] \end{align*} where :math:`N` is the number of datapoints, :math:`q(\mathbf u)` is the variational distribution for the inducing function values, :math:`q(f_i)` is the marginal of :math:`p(f_i \mid \mathbf u, \mathbf x_i) q(\mathbf u)`, and :math:`p(\mathbf u)` is the prior distribution for the inducing function values. :math:`\beta` is a scaling constant that reduces the regularization effect of the KL divergence. Setting :math:`\beta=1` (default) results in the true variational ELBO. For more information on this derivation, see `Scalable Variational Gaussian Process Classification`_ (Hensman et al., 2015). :param ~gpytorch.likelihoods.Likelihood likelihood: The likelihood for the model :param ~gpytorch.models.ApproximateGP model: The approximate GP model :param int num_data: The total number of training data points (necessary for SGD) :param float beta: (optional, default=1.) A multiplicative factor for the KL divergence term. Setting it to 1 (default) recovers true variational inference (as derived in `Scalable Variational Gaussian Process Classification`_). Setting it to anything less than 1 reduces the regularization effect of the model (similarly to what was proposed in `the beta-VAE paper`_). :param bool combine_terms: (default=True): Whether or not to sum the expected NLL with the KL terms (default True) Example: >>> # model is a gpytorch.models.ApproximateGP >>> # likelihood is a gpytorch.likelihoods.Likelihood >>> mll = gpytorch.mlls.VariationalELBO(likelihood, model, num_data=100, beta=0.5) >>> >>> output = model(train_x) >>> loss = -mll(output, train_y) >>> loss.backward() .. _Scalable Variational Gaussian Process Classification: .. _the beta-VAE paper: """ def _log_likelihood_term(self, variational_dist_f, target, **kwargs): return self.likelihood.expected_log_prob(target, variational_dist_f, **kwargs).sum(-1)
[docs] def forward(self, variational_dist_f, target, **kwargs): r""" Computes the Variational ELBO given :math:`q(\mathbf f)` and :math:`\mathbf y`. Calling this function will call the likelihood's :meth:`~gpytorch.likelihoods.Likelihood.expected_log_prob` function. :param ~gpytorch.distributions.MultivariateNormal variational_dist_f: :math:`q(\mathbf f)` the outputs of the latent function (the :obj:`gpytorch.models.ApproximateGP`) :param torch.Tensor target: :math:`\mathbf y` The target values :param kwargs: Additional arguments passed to the likelihood's :meth:`~gpytorch.likelihoods.Likelihood.expected_log_prob` function. :rtype: torch.Tensor :return: Variational ELBO. Output shape corresponds to batch shape of the model/input data. """ return super().forward(variational_dist_f, target, **kwargs)