#!/usr/bin/env python3
# Copyright (c) Facebook, Inc. and its affiliates.
#
# This source code is licensed under the MIT license found in the
# LICENSE file in the root directory of this source tree.
from typing import Optional
import torch
from botorch.posteriors.gpytorch import GPyTorchPosterior
from gpytorch.distributions import MultivariateNormal
from gpytorch.lazy import LazyTensor
from torch import Tensor
[docs]class HigherOrderGPPosterior(GPyTorchPosterior):
r"""
Posterior class for a Higher order Gaussian process model [Zhe2019hogp]_. Extends
the standard GPyTorch posterior class by overwriting the rsample method.
The posterior variance is handled internally by the HigherOrderGP model.
HOGP is a tensorized GP model so the posterior covariance grows to be extremely
large, but is highly structured, which means that we can exploit Kronecker
identities to sample from the posterior using Matheron's rule as described in
[Doucet2010sampl]_.
In general, this posterior should ONLY be used for HOGP models
that have highly structured covariances. It should also only be used internally when
called from the HigherOrderGP.posterior(...) method. At this time, the posterior
does not support gradients with respect to the training data.
"""
def __init__(
self,
mvn: MultivariateNormal,
joint_covariance_matrix: LazyTensor,
train_train_covar: LazyTensor,
test_train_covar: LazyTensor,
train_targets: Tensor,
output_shape: torch.Size,
num_outputs: int,
) -> None:
r"""A Posterior for HigherOrderGP models.
Args:
mvn: Posterior multivariate normal distribution
joint_covariance_matrix: Joint test train covariance matrix over the entire
tensor
train_train_covar: covariance matrix of train points in the data space
test_train_covar: covariance matrix of test x train points in the data space
train_targets: training responses vectorized
output_shape: shape output training responses
num_outputs: batch shaping of model
"""
super().__init__(mvn)
self.joint_covariance_matrix = joint_covariance_matrix
self.train_train_covar = train_train_covar
self.test_train_covar = test_train_covar
self.train_targets = train_targets
self.output_shape = output_shape
self._is_mt = True
self.num_outputs = num_outputs
@property
def base_sample_shape(self):
# overwrites the standard base_sample_shape call to inform samplers that
# n + 2 n_train samples need to be drawn rather than n samples
joint_covar = self.joint_covariance_matrix
batch_shape = joint_covar.shape[:-2]
sampling_shape = torch.Size(
[joint_covar.shape[-2] + self.train_train_covar.shape[-2]]
)
return batch_shape + sampling_shape
@property
def event_shape(self):
return self.output_shape
def _prepare_base_samples(
self, sample_shape: torch.Size, base_samples: Tensor = None
) -> Tensor:
covariance_matrix = self.joint_covariance_matrix
joint_size = covariance_matrix.shape[-1]
batch_shape = covariance_matrix.batch_shape
if base_samples is not None:
if base_samples.shape[: len(sample_shape)] != sample_shape:
raise RuntimeError("sample_shape disagrees with shape of base_samples.")
appended_shape = joint_size + self.train_train_covar.shape[-1]
if appended_shape != base_samples.shape[-1]:
# get base_samples to the correct shape by expanding as sample shape,
# batch shape, then rest of dimensions. We have to add first the sample
# shape, then the batch shape of the model, and then finally the shape
# of the test data points squeezed into a single dimension, accessed
# from the test_train_covar.
base_sample_shapes = (
sample_shape + batch_shape + self.test_train_covar.shape[-2:-1]
)
if base_samples.nelement() == base_sample_shapes.numel():
base_samples = base_samples.reshape(base_sample_shapes)
new_base_samples = torch.randn(
*sample_shape,
*batch_shape,
appended_shape - base_samples.shape[-1],
device=base_samples.device,
dtype=base_samples.dtype,
)
base_samples = torch.cat((base_samples, new_base_samples), dim=-1)
else:
# nuke the base samples if we cannot use them.
base_samples = None
if base_samples is None:
# TODO: Allow qMC sampling
base_samples = torch.randn(
*sample_shape,
*batch_shape,
joint_size,
device=covariance_matrix.device,
dtype=covariance_matrix.dtype,
)
noise_base_samples = torch.randn(
*sample_shape,
*batch_shape,
self.train_train_covar.shape[-1],
device=covariance_matrix.device,
dtype=covariance_matrix.dtype,
)
else:
# finally split up the base samples
noise_base_samples = base_samples[..., joint_size:]
base_samples = base_samples[..., :joint_size]
perm_list = [*range(1, base_samples.ndim), 0]
return base_samples.permute(*perm_list), noise_base_samples.permute(*perm_list)
[docs] def rsample(
self,
sample_shape: Optional[torch.Size] = None,
base_samples: Optional[Tensor] = None,
) -> Tensor:
r"""Sample from the posterior (with gradients).
As the posterior covariance is difficult to draw from in this model,
we implement Matheron's rule as described in [Doucet2010sampl]-. This may not
work entirely correctly for deterministic base samples unless base samples
are provided that are of shape `n + 2 * n_train` because the sampling method
draws `2 * n_train` samples as well as the standard `n`.
samples.
Args:
sample_shape: A `torch.Size` object specifying the sample shape. To
draw `n` samples, set to `torch.Size([n])`. To draw `b` batches
of `n` samples each, set to `torch.Size([b, n])`.
base_samples: An (optional) Tensor of `N(0, I)` base samples of
appropriate dimension, typically obtained from a `Sampler`.
This is used for deterministic optimization.
Returns:
A `sample_shape x event_shape`-dim Tensor of samples from the posterior.
"""
if sample_shape is None:
sample_shape = torch.Size([1])
base_samples, noise_base_samples = self._prepare_base_samples(
sample_shape, base_samples
)
# base samples now have trailing sample dimension
covariance_matrix = self.joint_covariance_matrix
covar_root = covariance_matrix.root_decomposition().root
samples = covar_root.matmul(base_samples[..., : covar_root.shape[-1], :])
# now pluck out Y_x and X_x
noiseless_train_marginal_samples = samples[
..., : self.train_train_covar.shape[-1], :
]
test_marginal_samples = samples[..., self.train_train_covar.shape[-1] :, :]
# we need to add noise to the train_joint_samples
# THIS ASSUMES CONSTANT NOISE
noise_std = self.train_train_covar.lazy_tensors[1]._diag[..., 0] ** 0.5
# TODO: cleanup the reshaping here
# expands the noise to allow broadcasting against the noise base samples
# reshape_as or view_as don't work here because we need to expand to
# broadcast against `samples x batch_shape x output_shape` while noise_std
# is `batch_shape x 1`.
if self.num_outputs > 1 or noise_std.ndim > 1:
ntms_dims = [
i == noise_std.shape[0] for i in noiseless_train_marginal_samples.shape
]
for matched in ntms_dims:
if not matched:
noise_std = noise_std.unsqueeze(-1)
# we need to add noise into the noiseless samples
noise_marginal_samples = noise_std * noise_base_samples
train_marginal_samples = (
noiseless_train_marginal_samples + noise_marginal_samples
)
# compute y - Y_x
train_rhs = self.train_targets - train_marginal_samples
# K_{train, train}^{-1} (y - Y_x)
# internally, this solve is done using Kronecker algebra and is fast.
kinv_rhs = self.train_train_covar.inv_matmul(train_rhs)
# multiply by cross-covariance
test_updated_samples = self.test_train_covar.matmul(kinv_rhs)
# add samples
test_cond_samples = test_marginal_samples + test_updated_samples
test_cond_samples = test_cond_samples.permute(
test_cond_samples.ndim - 1, *range(0, test_cond_samples.ndim - 1)
)
# reshape samples to be the actual size of the train targets
return test_cond_samples.reshape(*sample_shape, *self.output_shape)
