Source code for

from __future__ import absolute_import, division, print_function

import torch
from torch.distributions import constraints
from torch.nn import Parameter

import pyro
import pyro.distributions as dist
from pyro.contrib import autoname
from import GPModel
from import conditional
from pyro.distributions.util import eye_like

[docs]class VariationalGP(GPModel): r""" Variational Gaussian Process model. This model deals with both Gaussian and non-Gaussian likelihoods. Given inputs\ :math:`X` and their noisy observations :math:`y`, the model takes the form .. math:: f &\sim \mathcal{GP}(0, k(X, X)),\\ y & \sim p(y) = p(y \mid f) p(f), where :math:`p(y \mid f)` is the likelihood. We will use a variational approach in this model by approximating :math:`q(f)` to the posterior :math:`p(f\mid y)`. Precisely, :math:`q(f)` will be a multivariate normal distribution with two parameters ``f_loc`` and ``f_scale_tril``, which will be learned during a variational inference process. .. note:: This model can be seen as a special version of :class:`.SparseVariationalGP` model with :math:`X_u = X`. .. note:: This model has :math:`\mathcal{O}(N^3)` complexity for training, :math:`\mathcal{O}(N^3)` complexity for testing. Here, :math:`N` is the number of train inputs. Size of variational parameters is :math:`\mathcal{O}(N^2)`. :param torch.Tensor X: A input data for training. Its first dimension is the number of data points. :param torch.Tensor y: An output data for training. Its last dimension is the number of data points. :param kernel: A Pyro kernel object, which is the covariance function :math:`k`. :param Likelihood likelihood: A likelihood object. :param callable mean_function: An optional mean function :math:`m` of this Gaussian process. By default, we use zero mean. :param torch.Size latent_shape: Shape for latent processes (`batch_shape` of :math:`q(f)`). By default, it equals to output batch shape ``y.shape[:-1]``. For the multi-class classification problems, ``latent_shape[-1]`` should corresponse to the number of classes. :param bool whiten: A flag to tell if variational parameters ``f_loc`` and ``f_scale_tril`` are transformed by the inverse of ``Lff``, where ``Lff`` is the lower triangular decomposition of :math:`kernel(X, X)`. Enable this flag will help optimization. :param float jitter: A small positive term which is added into the diagonal part of a covariance matrix to help stablize its Cholesky decomposition. """ def __init__(self, X, y, kernel, likelihood, mean_function=None, latent_shape=None, whiten=False, jitter=1e-6): super(VariationalGP, self).__init__(X, y, kernel, mean_function, jitter) self.likelihood = likelihood y_batch_shape = self.y.shape[:-1] if self.y is not None else torch.Size([]) self.latent_shape = latent_shape if latent_shape is not None else y_batch_shape N = self.X.size(0) f_loc = self.X.new_zeros(self.latent_shape + (N,)) self.f_loc = Parameter(f_loc) identity = eye_like(self.X, N) f_scale_tril = identity.repeat(self.latent_shape + (1, 1)) self.f_scale_tril = Parameter(f_scale_tril) self.set_constraint("f_scale_tril", constraints.lower_cholesky) self.whiten = whiten self._sample_latent = True
[docs] @autoname.scope(prefix="VGP") def model(self): self.set_mode("model") N = self.X.size(0) Kff = self.kernel(self.X).contiguous() Kff.view(-1)[::N + 1] += self.jitter # add jitter to the diagonal Lff = Kff.cholesky() zero_loc = self.X.new_zeros(self.f_loc.shape) if self.whiten: identity = eye_like(self.X, N) pyro.sample("f", dist.MultivariateNormal(zero_loc, scale_tril=identity) .to_event(zero_loc.dim() - 1)) f_scale_tril = Lff.matmul(self.f_scale_tril) f_loc = Lff.matmul(self.f_loc.unsqueeze(-1)).squeeze(-1) else: pyro.sample("f", dist.MultivariateNormal(zero_loc, scale_tril=Lff) .to_event(zero_loc.dim() - 1)) f_scale_tril = self.f_scale_tril f_loc = self.f_loc f_loc = f_loc + self.mean_function(self.X) f_var = f_scale_tril.pow(2).sum(dim=-1) if self.y is None: return f_loc, f_var else: return self.likelihood(f_loc, f_var, self.y)
[docs] @autoname.scope(prefix="VGP") def guide(self): self.set_mode("guide") pyro.sample("f", dist.MultivariateNormal(self.f_loc, scale_tril=self.f_scale_tril) .to_event(self.f_loc.dim()-1))
[docs] def forward(self, Xnew, full_cov=False): r""" Computes the mean and covariance matrix (or variance) of Gaussian Process posterior on a test input data :math:`X_{new}`: .. math:: p(f^* \mid X_{new}, X, y, k, f_{loc}, f_{scale\_tril}) = \mathcal{N}(loc, cov). .. note:: Variational parameters ``f_loc``, ``f_scale_tril``, together with kernel's parameters have been learned from a training procedure (MCMC or SVI). :param torch.Tensor Xnew: A input data for testing. Note that ``Xnew.shape[1:]`` must be the same as ``self.X.shape[1:]``. :param bool full_cov: A flag to decide if we want to predict full covariance matrix or just variance. :returns: loc and covariance matrix (or variance) of :math:`p(f^*(X_{new}))` :rtype: tuple(torch.Tensor, torch.Tensor) """ self._check_Xnew_shape(Xnew) self.set_mode("guide") loc, cov = conditional(Xnew, self.X, self.kernel, self.f_loc, self.f_scale_tril, full_cov=full_cov, whiten=self.whiten, jitter=self.jitter) return loc + self.mean_function(Xnew), cov