#!/usr/bin/env python3
# Copyright (c) Meta Platforms, Inc. and affiliates.
#
# This source code is licensed under the MIT license found in the
# LICENSE file in the root directory of this source tree.
r"""
References
.. [Zhe2019hogp]
S. Zhe, W. Xing, and R. M. Kirby. Scalable high-order gaussian process regression.
Proceedings of Machine Learning Research, volume 89, Apr 2019.
"""
from __future__ import annotations
import warnings
from contextlib import ExitStack
from typing import Any, List, Optional, Tuple, Union
import torch
from botorch.acquisition.objective import PosteriorTransform
from botorch.models.gpytorch import BatchedMultiOutputGPyTorchModel
from botorch.models.model import FantasizeMixin
from botorch.models.transforms.input import InputTransform
from botorch.models.transforms.outcome import OutcomeTransform, Standardize
from botorch.models.utils import gpt_posterior_settings
from botorch.posteriors import (
GPyTorchPosterior,
HigherOrderGPPosterior,
TransformedPosterior,
)
from gpytorch.constraints import GreaterThan
from gpytorch.distributions import MultivariateNormal
from gpytorch.kernels import Kernel, MaternKernel
from gpytorch.likelihoods import GaussianLikelihood, Likelihood
from gpytorch.models import ExactGP
from gpytorch.priors.torch_priors import GammaPrior, MultivariateNormalPrior
from gpytorch.settings import fast_pred_var, skip_posterior_variances
from linear_operator.operators import (
BatchRepeatLinearOperator,
DiagLinearOperator,
KroneckerProductLinearOperator,
LinearOperator,
ZeroLinearOperator,
)
from torch import Tensor
from torch.nn import ModuleList, Parameter, ParameterList
MIN_INFERRED_NOISE_LEVEL = 1e-4
[docs]class FlattenedStandardize(Standardize):
r"""
Standardize outcomes in a structured multi-output settings by reshaping the
batched output dimensions to be a vector. Specifically, an output dimension
of [a x b x c] will be squeezed to be a vector of [a * b * c].
"""
def __init__(
self,
output_shape: torch.Size,
batch_shape: torch.Size = None,
min_stdv: float = 1e-8,
):
r"""
Args:
output_shape: A `n x output_shape`-dim tensor of training targets.
batch_shape: The batch_shape of the training targets.
min_stddv: The minimum standard deviation for which to perform
standardization (if lower, only de-mean the data).
"""
if batch_shape is None:
batch_shape = torch.Size()
super(FlattenedStandardize, self).__init__(
m=1, outputs=None, batch_shape=batch_shape, min_stdv=min_stdv
)
self.output_shape = output_shape
self.batch_shape = batch_shape
def _squeeze_to_single_output(self, tsr: Tensor) -> Tensor:
dim_ct = tsr.ndim - len(self.output_shape) - 1
return tsr.reshape(*tsr.shape[:dim_ct], -1, 1)
def _return_to_output_shape(self, tsr: Tensor) -> Tensor:
out = tsr.reshape(*tsr.shape[:-2], -1, *self.output_shape)
return out
[docs] def forward(
self, Y: Tensor, Yvar: Optional[Tensor] = None
) -> Tuple[Tensor, Optional[Tensor]]:
Y = self._squeeze_to_single_output(Y)
if Yvar is not None:
Yvar = self._squeeze_to_single_output(Yvar)
Y, Yvar = super().forward(Y, Yvar)
Y_out = self._return_to_output_shape(Y)
if Yvar is not None:
Yvar_out = self._return_to_output_shape(Yvar)
else:
Yvar_out = None
return Y_out, Yvar_out
[docs] def untransform_posterior(
self, posterior: HigherOrderGPPosterior
) -> TransformedPosterior:
# TODO: return a HigherOrderGPPosterior once rescaling constant
# muls * LinearOperators won't force a dense decomposition rather than a
# Kronecker structured one.
return TransformedPosterior(
posterior=posterior,
sample_transform=lambda s: self._return_to_output_shape(
self.means + self.stdvs * self._squeeze_to_single_output(s)
),
mean_transform=lambda m, v: self._return_to_output_shape(
self.means + self.stdvs * self._squeeze_to_single_output(m)
),
variance_transform=lambda m, v: self._return_to_output_shape(
self._stdvs_sq * self._squeeze_to_single_output(v)
),
)
[docs]class HigherOrderGP(BatchedMultiOutputGPyTorchModel, ExactGP, FantasizeMixin):
r"""
A model for high-dimensional output regression.
As described in [Zhe2019hogp]_. “Higher-order” means that the predictions
are matrices (tensors) with at least two dimensions, such as images or
grids of images, or measurements taken from a region of at least two
dimensions.
The posterior uses Matheron's rule [Doucet2010sampl]_
as described in [Maddox2021bohdo]_.
`HigherOrderGP` differs from a "vector” multi-output model in that it uses
Kronecker algebra to obtain parsimonious covariance matrices for these
outputs (see `KroneckerMultiTaskGP` for more information). For example,
imagine a 10 x 20 x 30 grid of images. If we were to vectorize the
resulting 6,000 data points in order to use them in a non-higher-order GP,
they would have a 6,000 x 6,000 covariance matrix, with 36 million entries.
The Kronecker structure allows representing this as a product of 10x10,
20x20, and 30x30 covariance matrices, with only 1,400 entries.
"""
def __init__(
self,
train_X: Tensor,
train_Y: Tensor,
likelihood: Optional[Likelihood] = None,
covar_modules: Optional[List[Kernel]] = None,
num_latent_dims: Optional[List[int]] = None,
learn_latent_pars: bool = True,
latent_init: str = "default",
outcome_transform: Optional[OutcomeTransform] = None,
input_transform: Optional[InputTransform] = None,
):
r"""
Args:
train_X: A `batch_shape x n x d`-dim tensor of training inputs.
train_Y: A `batch_shape x n x output_shape`-dim tensor of training targets.
likelihood: Gaussian likelihood for the model.
covar_modules: List of kernels for each output structure.
num_latent_dims: Sizes for the latent dimensions.
learn_latent_pars: If true, learn the latent parameters.
latent_init: [default or gp] how to initialize the latent parameters.
"""
if input_transform is not None:
input_transform.to(train_X)
# infer the dimension of `output_shape`.
num_output_dims = train_Y.dim() - train_X.dim() + 1
batch_shape = train_X.shape[:-2]
if len(batch_shape) > 1:
raise NotImplementedError(
"HigherOrderGP currently only supports 1-dim `batch_shape`."
)
if outcome_transform is not None:
if isinstance(outcome_transform, Standardize) and not isinstance(
outcome_transform, FlattenedStandardize
):
warnings.warn(
"HigherOrderGP does not support the outcome_transform "
"`Standardize`! Using `FlattenedStandardize` with `output_shape="
f"{train_Y.shape[- num_output_dims:]} and batch_shape="
f"{batch_shape} instead.",
RuntimeWarning,
)
outcome_transform = FlattenedStandardize(
output_shape=train_Y.shape[-num_output_dims:],
batch_shape=batch_shape,
)
train_Y, _ = outcome_transform(train_Y)
self._aug_batch_shape = batch_shape
self._num_dimensions = num_output_dims + 1
self._num_outputs = train_Y.shape[0] if batch_shape else 1
self.target_shape = train_Y.shape[-num_output_dims:]
self._input_batch_shape = batch_shape
if likelihood is None:
noise_prior = GammaPrior(1.1, 0.05)
noise_prior_mode = (noise_prior.concentration - 1) / noise_prior.rate
likelihood = GaussianLikelihood(
noise_prior=noise_prior,
batch_shape=self._aug_batch_shape,
noise_constraint=GreaterThan(
MIN_INFERRED_NOISE_LEVEL,
transform=None,
initial_value=noise_prior_mode,
),
)
else:
self._is_custom_likelihood = True
super().__init__(
train_X,
train_Y.view(*self._aug_batch_shape, -1),
likelihood=likelihood,
)
if covar_modules is not None:
self.covar_modules = ModuleList(covar_modules)
else:
self.covar_modules = ModuleList(
[
MaternKernel(
nu=2.5,
lengthscale_prior=GammaPrior(3.0, 6.0),
batch_shape=self._aug_batch_shape,
ard_num_dims=1 if dim > 0 else train_X.shape[-1],
)
for dim in range(self._num_dimensions)
]
)
if num_latent_dims is None:
num_latent_dims = [1] * (self._num_dimensions - 1)
self.to(train_X)
self._initialize_latents(
latent_init=latent_init,
num_latent_dims=num_latent_dims,
learn_latent_pars=learn_latent_pars,
device=train_Y.device,
dtype=train_Y.dtype,
)
if outcome_transform is not None:
self.outcome_transform = outcome_transform
if input_transform is not None:
self.input_transform = input_transform
def _initialize_latents(
self,
latent_init: str,
num_latent_dims: List[int],
learn_latent_pars: bool,
device: torch.device,
dtype: torch.dtype,
):
self.latent_parameters = ParameterList()
if latent_init == "default":
for dim_num in range(len(self.covar_modules) - 1):
self.latent_parameters.append(
Parameter(
torch.rand(
*self._aug_batch_shape,
self.target_shape[dim_num],
num_latent_dims[dim_num],
device=device,
dtype=dtype,
),
requires_grad=learn_latent_pars,
)
)
elif latent_init == "gp":
for dim_num, covar in enumerate(self.covar_modules[1:]):
latent_covar = covar(
torch.linspace(
0.0,
1.0,
self.target_shape[dim_num],
device=device,
dtype=dtype,
)
).add_jitter(1e-4)
latent_dist = MultivariateNormal(
torch.zeros(
*self._aug_batch_shape,
self.target_shape[dim_num],
device=device,
dtype=dtype,
),
latent_covar,
)
sample_shape = torch.Size((num_latent_dims[dim_num],))
latent_sample = latent_dist.sample(sample_shape=sample_shape)
latent_sample = latent_sample.reshape(
*self._aug_batch_shape,
self.target_shape[dim_num],
num_latent_dims[dim_num],
)
self.latent_parameters.append(
Parameter(
latent_sample,
requires_grad=learn_latent_pars,
)
)
self.register_prior(
"latent_parameters_" + str(dim_num),
MultivariateNormalPrior(
latent_dist.loc,
latent_dist.covariance_matrix.detach().clone(),
transform=lambda x: x.squeeze(-1),
),
lambda module, dim_num=dim_num: self.latent_parameters[dim_num],
)
[docs] def forward(self, X: Tensor) -> MultivariateNormal:
if self.training:
X = self.transform_inputs(X)
covariance_list = []
covariance_list.append(self.covar_modules[0](X))
for cm, param in zip(self.covar_modules[1:], self.latent_parameters):
if not self.training:
with torch.no_grad():
covariance_list.append(cm(param))
else:
covariance_list.append(cm(param))
# check batch_shapes
if covariance_list[0].batch_shape != covariance_list[1].batch_shape:
for i in range(1, len(covariance_list)):
cm = covariance_list[i]
covariance_list[i] = BatchRepeatLinearOperator(
cm, covariance_list[0].batch_shape
)
kronecker_covariance = KroneckerProductLinearOperator(*covariance_list)
# TODO: expand options for the mean module via batch shaping?
mean = torch.zeros(
*covariance_list[0].batch_shape,
kronecker_covariance.shape[-1],
device=kronecker_covariance.device,
dtype=kronecker_covariance.dtype,
)
return MultivariateNormal(mean, kronecker_covariance)
[docs] def get_fantasy_model(self, inputs, targets, **kwargs):
# we need to squeeze the targets in order to preserve the shaping
inputs_batch_dims = len(inputs.shape[:-2])
target_shape = (*inputs.shape[:-2], -1)
if (inputs_batch_dims + self._num_dimensions) < targets.ndim:
target_shape = (targets.shape[0], *target_shape)
reshaped_targets = targets.view(*target_shape)
return super().get_fantasy_model(inputs, reshaped_targets, **kwargs)
[docs] def condition_on_observations(
self, X: Tensor, Y: Tensor, **kwargs: Any
) -> HigherOrderGP:
r"""Condition the model on new observations.
Args:
X: A `batch_shape x n' x d`-dim Tensor, where `d` is the dimension of
the feature space, `m` is the number of points per batch, and
`batch_shape` is the batch shape (must be compatible with the
batch shape of the model).
Y: A `batch_shape' x n' x m_d`-dim Tensor, where `m_d` is the shaping
of the model outputs, `n'` is the number of points per batch, and
`batch_shape'` is the batch shape of the observations.
`batch_shape'` must be broadcastable to `batch_shape` using
standard broadcasting semantics. If `Y` has fewer batch dimensions
than `X`, its is assumed that the missing batch dimensions are
the same for all `Y`.
Returns:
A `BatchedMultiOutputGPyTorchModel` object of the same type with
`n + n'` training examples, representing the original model
conditioned on the new observations `(X, Y)` (and possibly noise
observations passed in via kwargs).
"""
noise = kwargs.get("noise")
if hasattr(self, "outcome_transform"):
# we need to apply transforms before shifting batch indices around
Y, noise = self.outcome_transform(Y, noise)
self._validate_tensor_args(X=X, Y=Y, Yvar=noise, strict=False)
# we don't need to do un-squeezing because Y already is batched
# we don't support fixed noise here yet
# if noise is not None:
# kwargs.update({"noise": noise})
fantasy_model = super(
BatchedMultiOutputGPyTorchModel, self
).condition_on_observations(X=X, Y=Y, **kwargs)
fantasy_model._input_batch_shape = fantasy_model.train_targets.shape[
: (-1 if self._num_outputs == 1 else -2)
]
fantasy_model._aug_batch_shape = fantasy_model.train_targets.shape[:-1]
return fantasy_model
[docs] def posterior(
self,
X: Tensor,
output_indices: Optional[List[int]] = None,
observation_noise: Union[bool, Tensor] = False,
posterior_transform: Optional[PosteriorTransform] = None,
**kwargs: Any,
) -> GPyTorchPosterior:
self.eval() # make sure we're calling a posterior
if posterior_transform is not None:
# this could be very costly, disallow for now
raise NotImplementedError(
"Posterior transforms currently not supported for "
f"{self.__class__.__name__}"
)
# input transforms are applied at `posterior` in `eval` mode, and at
# `model.forward()` at the training time
X = self.transform_inputs(X)
no_pred_variance = skip_posterior_variances._state
with ExitStack() as es:
es.enter_context(gpt_posterior_settings())
es.enter_context(fast_pred_var(True))
# we need to skip posterior variances here
es.enter_context(skip_posterior_variances(True))
mvn = self(X)
if observation_noise is not False:
# TODO: ensure that this still works for structured noise solves.
mvn = self.likelihood(mvn, X)
# lazy covariance matrix includes the interpolated version of the full
# covariance matrix so we can actually grab that instead.
if X.ndimension() > self.train_inputs[0].ndimension():
X_batch_shape = X.shape[:-2]
train_inputs = self.train_inputs[0].reshape(
*[1] * len(X_batch_shape), *self.train_inputs[0].shape
)
train_inputs = train_inputs.repeat(
*X_batch_shape, *[1] * self.train_inputs[0].ndimension()
)
else:
train_inputs = self.train_inputs[0]
# we now compute the data covariances for the training data, the testing
# data, the joint covariances, and the test train cross-covariance
train_train_covar = self.prediction_strategy.lik_train_train_covar.detach()
base_train_train_covar = train_train_covar.lazy_tensor
data_train_covar = base_train_train_covar.linear_ops[0]
data_covar = self.covar_modules[0]
data_train_test_covar = data_covar(X, train_inputs)
data_test_test_covar = data_covar(X)
data_joint_covar = data_train_covar.cat_rows(
cross_mat=data_train_test_covar,
new_mat=data_test_test_covar,
)
# we detach the latents so that they don't cause gradient errors
# TODO: Can we enable backprop through the latent covariances?
batch_shape = data_train_test_covar.batch_shape
latent_covar_list = []
for latent_covar in base_train_train_covar.linear_ops[1:]:
if latent_covar.batch_shape != batch_shape:
latent_covar = BatchRepeatLinearOperator(latent_covar, batch_shape)
latent_covar_list.append(latent_covar.detach())
joint_covar = KroneckerProductLinearOperator(
data_joint_covar, *latent_covar_list
)
test_train_covar = KroneckerProductLinearOperator(
data_train_test_covar, *latent_covar_list
)
# compute the posterior variance if necessary
if no_pred_variance:
pred_variance = mvn.variance
else:
pred_variance = self.make_posterior_variances(joint_covar)
# mean and variance get reshaped into the target shape
new_mean = mvn.mean.reshape(*X.shape[:-1], *self.target_shape)
if not no_pred_variance:
new_variance = pred_variance.reshape(*X.shape[:-1], *self.target_shape)
new_variance = DiagLinearOperator(new_variance)
else:
new_variance = ZeroLinearOperator(
*X.shape[:-1], *self.target_shape, self.target_shape[-1]
)
mvn = MultivariateNormal(new_mean, new_variance)
# return a specialized Posterior to allow for sampling
# cloning the full covar allows backpropagation through it
posterior = HigherOrderGPPosterior(
mvn=mvn,
train_targets=self.train_targets.unsqueeze(-1),
train_train_covar=train_train_covar,
test_train_covar=test_train_covar,
joint_covariance_matrix=joint_covar.clone(),
output_shape=X.shape[:-1] + self.target_shape,
num_outputs=self._num_outputs,
)
if hasattr(self, "outcome_transform"):
posterior = self.outcome_transform.untransform_posterior(posterior)
return posterior
[docs] def make_posterior_variances(
self, joint_covariance_matrix: LinearOperator
) -> Tensor:
r"""
Computes the posterior variances given the data points X. As currently
implemented, it computes another forwards call with the stacked data to get out
the joint covariance across all data points.
"""
# TODO: use the exposed joint covariances from the prediction strategy
data_joint_covariance = joint_covariance_matrix.linear_ops[0].evaluate_kernel()
num_train = self.train_inputs[0].shape[-2]
test_train_covar = data_joint_covariance[..., num_train:, :num_train]
train_train_covar = data_joint_covariance[..., :num_train, :num_train]
test_test_covar = data_joint_covariance[..., num_train:, num_train:]
jcm_linops = joint_covariance_matrix.linear_ops[1:]
full_train_train_covar = KroneckerProductLinearOperator(
train_train_covar, *jcm_linops
)
full_test_test_covar = KroneckerProductLinearOperator(
test_test_covar, *jcm_linops
)
full_test_train_covar_tuple = (test_train_covar,) + jcm_linops
train_evals, train_evecs = full_train_train_covar.symeig(eigenvectors=True)
# (\kron \Lambda_i + \sigma^2 I)^{-1}
train_inv_evals = DiagLinearOperator(
1.0 / (train_evals + self.likelihood.noise)
)
# compute K_i S_i \hadamard K_i S_i
test_train_hadamard = KroneckerProductLinearOperator(
*[
lt1.matmul(lt2).to_dense() ** 2
for lt1, lt2 in zip(full_test_train_covar_tuple, train_evecs.linear_ops)
]
)
# and compute the column sums of
# (\kron K_i S_i * K_i S_i) \tilde{\Lambda}^{-1}
test_train_pred_covar = test_train_hadamard.matmul(train_inv_evals).sum(dim=-1)
pred_variances = full_test_test_covar.diag() - test_train_pred_covar
return pred_variances
