#!/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:
http://proceedings.mlr.press/v38/hensman15.pdf
.. _the beta-VAE paper:
https://openreview.net/pdf?id=Sy2fzU9gl
"""
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)