#!/usr/bin/env python3
from typing import Any, Optional, Union
import torch
from linear_operator import to_linear_operator
from linear_operator.operators import (
ConstantDiagLinearOperator,
DiagLinearOperator,
KroneckerProductDiagLinearOperator,
KroneckerProductLinearOperator,
LinearOperator,
RootLinearOperator,
)
from torch import Tensor
from torch.distributions import Normal
from ..constraints import GreaterThan, Interval
from ..distributions import base_distributions, MultitaskMultivariateNormal
from ..lazy import LazyEvaluatedKernelTensor
from ..likelihoods import _GaussianLikelihoodBase, Likelihood
from ..priors import Prior
from .noise_models import FixedGaussianNoise, Noise
class _MultitaskGaussianLikelihoodBase(_GaussianLikelihoodBase):
r"""
Base class for multi-task Gaussian Likelihoods, supporting general heteroskedastic noise models.
:param num_tasks: Number of tasks.
:param noise_covar: A model for the noise covariance. This can be a simple homoskedastic noise model, or a GP
that is to be fitted on the observed measurement errors.
:param rank: The rank of the task noise covariance matrix to fit. If `rank`
is set to 0, then a diagonal covariance matrix is fit.
:param task_correlation_prior: Prior to use over the task noise correlation
matrix. Only used when :math:`\text{rank} > 0`.
:param batch_shape: Number of batches.
"""
def __init__(
self,
num_tasks: int,
noise_covar: Union[Noise, FixedGaussianNoise],
rank: int = 0,
task_correlation_prior: Optional[Prior] = None,
batch_shape: torch.Size = torch.Size(),
) -> None:
super().__init__(noise_covar=noise_covar)
if rank != 0:
if rank > num_tasks:
raise ValueError(f"Cannot have rank ({rank}) greater than num_tasks ({num_tasks})")
tidcs = torch.tril_indices(num_tasks, rank, dtype=torch.long)
self.tidcs: Tensor = tidcs[:, 1:] # (1, 1) must be 1.0, no need to parameterize this
task_noise_corr = torch.randn(*batch_shape, self.tidcs.size(-1))
self.register_parameter("task_noise_corr", torch.nn.Parameter(task_noise_corr))
if task_correlation_prior is not None:
self.register_prior(
"MultitaskErrorCorrelationPrior", task_correlation_prior, lambda m: m._eval_corr_matrix
)
elif task_correlation_prior is not None:
raise ValueError("Can only specify task_correlation_prior if rank>0")
self.num_tasks = num_tasks
self.rank = rank
def _eval_corr_matrix(self) -> Tensor:
tnc = self.task_noise_corr
fac_diag = torch.ones(*tnc.shape[:-1], self.num_tasks, device=tnc.device, dtype=tnc.dtype)
Cfac = torch.diag_embed(fac_diag)
Cfac[..., self.tidcs[0], self.tidcs[1]] = self.task_noise_corr
# squared rows must sum to one for this to be a correlation matrix
C = Cfac / Cfac.pow(2).sum(dim=-1, keepdim=True).sqrt()
return C @ C.transpose(-1, -2)
def marginal(
self, function_dist: MultitaskMultivariateNormal, *params: Any, **kwargs: Any
) -> MultitaskMultivariateNormal: # pyre-ignore[14]
r"""
If :math:`\text{rank} = 0`, adds the task noises to the diagonal of the
covariance matrix of the supplied
:obj:`~gpytorch.distributions.MultivariateNormal` or
:obj:`~gpytorch.distributions.MultitaskMultivariateNormal`. Otherwise,
adds a rank `rank` covariance matrix to it.
To accomplish this, we form a new
:obj:`~linear_operator.operators.KroneckerProductLinearOperator`
between :math:`I_{n}`, an identity matrix with size equal to the data
and a (not necessarily diagonal) matrix containing the task noises
:math:`D_{t}`.
We also incorporate a shared `noise` parameter from the base
:class:`gpytorch.likelihoods.GaussianLikelihood` that we extend.
The final covariance matrix after this method is then
:math:`\mathbf K + \mathbf D_{t} \otimes \mathbf I_{n} + \sigma^{2} \mathbf I_{nt}`.
:param function_dist: Random variable whose covariance
matrix is a :obj:`~linear_operator.operators.LinearOperator` we intend to augment.
:rtype: `gpytorch.distributions.MultitaskMultivariateNormal`:
:return: A new random variable whose covariance matrix is a
:obj:`~linear_operator.operators.LinearOperator` with
:math:`\mathbf D_{t} \otimes \mathbf I_{n}` and :math:`\sigma^{2} \mathbf I_{nt}` added.
"""
mean, covar = function_dist.mean, function_dist.lazy_covariance_matrix
# ensure that sumKroneckerLT is actually called
if isinstance(covar, LazyEvaluatedKernelTensor):
covar = covar.evaluate_kernel()
covar_kron_lt = self._shaped_noise_covar(
mean.shape, add_noise=self.has_global_noise, interleaved=function_dist._interleaved
)
covar = covar + covar_kron_lt
return function_dist.__class__(mean, covar, interleaved=function_dist._interleaved)
def _shaped_noise_covar(
self, shape: torch.Size, add_noise: Optional[bool] = True, interleaved: bool = True, *params: Any, **kwargs: Any
) -> LinearOperator:
if not self.has_task_noise:
noise = ConstantDiagLinearOperator(self.noise, diag_shape=shape[-2] * self.num_tasks)
return noise
if self.rank == 0:
task_noises = self.raw_task_noises_constraint.transform(self.raw_task_noises)
task_var_lt = DiagLinearOperator(task_noises)
dtype, device = task_noises.dtype, task_noises.device
ckl_init = KroneckerProductDiagLinearOperator
else:
task_noise_covar_factor = self.task_noise_covar_factor
task_var_lt = RootLinearOperator(task_noise_covar_factor)
dtype, device = task_noise_covar_factor.dtype, task_noise_covar_factor.device
ckl_init = KroneckerProductLinearOperator
eye_lt = ConstantDiagLinearOperator(
torch.ones(*shape[:-2], 1, dtype=dtype, device=device), diag_shape=shape[-2]
)
task_var_lt = task_var_lt.expand(*shape[:-2], *task_var_lt.matrix_shape) # pyre-ignore[6]
# to add the latent noise we exploit the fact that
# I \kron D_T + \sigma^2 I_{NT} = I \kron (D_T + \sigma^2 I)
# which allows us to move the latent noise inside the task dependent noise
# thereby allowing exploitation of Kronecker structure in this likelihood.
if add_noise and self.has_global_noise:
noise = ConstantDiagLinearOperator(self.noise, diag_shape=task_var_lt.shape[-1])
task_var_lt = task_var_lt + noise
if interleaved:
covar_kron_lt = ckl_init(eye_lt, task_var_lt)
else:
covar_kron_lt = ckl_init(task_var_lt, eye_lt)
return covar_kron_lt
def forward(self, function_samples: Tensor, *params: Any, **kwargs: Any) -> Normal:
noise = self._shaped_noise_covar(function_samples.shape, *params, **kwargs).diagonal(dim1=-1, dim2=-2)
noise = noise.reshape(*noise.shape[:-1], *function_samples.shape[-2:])
return base_distributions.Independent(base_distributions.Normal(function_samples, noise.sqrt()), 1)
[docs]class MultitaskGaussianLikelihood(_MultitaskGaussianLikelihoodBase):
r"""
A convenient extension of the :class:`~gpytorch.likelihoods.GaussianLikelihood` to the multitask setting that allows
for a full cross-task covariance structure for the noise. The fitted covariance matrix has rank `rank`.
If a strictly diagonal task noise covariance matrix is desired, then rank=0 should be set. (This option still
allows for a different `noise` parameter for each task.)
Like the Gaussian likelihood, this object can be used with exact inference.
.. note::
At least one of :attr:`has_global_noise` or :attr:`has_task_noise` should be specified.
.. note::
MultittaskGaussianLikelihood has an analytic marginal distribution.
:param num_tasks: Number of tasks.
:param noise_covar: A model for the noise covariance. This can be a simple homoskedastic noise model, or a GP
that is to be fitted on the observed measurement errors.
:param rank: The rank of the task noise covariance matrix to fit. If `rank`
is set to 0, then a diagonal covariance matrix is fit.
:param task_prior: Prior to use over the task noise correlation
matrix. Only used when :math:`\text{rank} > 0`.
:param batch_shape: Number of batches.
:param has_global_noise: Whether to include a :math:`\sigma^2 \mathbf I_{nt}` term in the noise model.
:param has_task_noise: Whether to include task-specific noise terms, which add
:math:`\mathbf I_n \otimes \mathbf D_T` into the noise model.
:ivar torch.Tensor task_noise_covar: The inter-task noise covariance matrix
:ivar torch.Tensor task_noises: (Optional) task specific noise variances (added onto the `task_noise_covar`)
:ivar torch.Tensor noise: (Optional) global noise variance (added onto the `task_noise_covar`)
"""
def __init__(
self,
num_tasks: int,
rank: int = 0,
batch_shape: torch.Size = torch.Size(),
task_prior: Optional[Prior] = None,
noise_prior: Optional[Prior] = None,
noise_constraint: Optional[Interval] = None,
has_global_noise: bool = True,
has_task_noise: bool = True,
) -> None:
super(Likelihood, self).__init__() # pyre-ignore[20]
if noise_constraint is None:
noise_constraint = GreaterThan(1e-4)
if not has_task_noise and not has_global_noise:
raise ValueError(
"At least one of has_task_noise or has_global_noise must be specified. "
"Attempting to specify a likelihood that has no noise terms."
)
if has_task_noise:
if rank == 0:
self.register_parameter(
name="raw_task_noises", parameter=torch.nn.Parameter(torch.zeros(*batch_shape, num_tasks))
)
self.register_constraint("raw_task_noises", noise_constraint)
if noise_prior is not None:
self.register_prior("raw_task_noises_prior", noise_prior, lambda m: m.task_noises)
if task_prior is not None:
raise RuntimeError("Cannot set a `task_prior` if rank=0")
else:
self.register_parameter(
name="task_noise_covar_factor",
parameter=torch.nn.Parameter(torch.randn(*batch_shape, num_tasks, rank)),
)
if task_prior is not None:
self.register_prior("MultitaskErrorCovariancePrior", task_prior, lambda m: m._eval_covar_matrix)
self.num_tasks = num_tasks
self.rank = rank
if has_global_noise:
self.register_parameter(name="raw_noise", parameter=torch.nn.Parameter(torch.zeros(*batch_shape, 1)))
self.register_constraint("raw_noise", noise_constraint)
if noise_prior is not None:
self.register_prior("raw_noise_prior", noise_prior, lambda m: m.noise)
self.has_global_noise = has_global_noise
self.has_task_noise = has_task_noise
@property
def noise(self) -> Optional[Tensor]:
return self.raw_noise_constraint.transform(self.raw_noise)
@noise.setter
def noise(self, value: Union[float, Tensor]) -> None:
self._set_noise(value)
@property
def task_noises(self) -> Optional[Tensor]:
if self.rank == 0:
return self.raw_task_noises_constraint.transform(self.raw_task_noises)
else:
raise AttributeError("Cannot set diagonal task noises when covariance has ", self.rank, ">0")
@task_noises.setter
def task_noises(self, value: Union[float, Tensor]) -> None:
if self.rank == 0:
self._set_task_noises(value)
else:
raise AttributeError("Cannot set diagonal task noises when covariance has ", self.rank, ">0")
def _set_noise(self, value: Union[float, Tensor]) -> None:
self.initialize(raw_noise=self.raw_noise_constraint.inverse_transform(value))
def _set_task_noises(self, value: Union[float, Tensor]) -> None:
self.initialize(raw_task_noises=self.raw_task_noises_constraint.inverse_transform(value))
@property
def task_noise_covar(self) -> Tensor:
if self.rank > 0:
return self.task_noise_covar_factor.matmul(self.task_noise_covar_factor.transpose(-1, -2))
else:
raise AttributeError("Cannot retrieve task noises when covariance is diagonal.")
@task_noise_covar.setter
def task_noise_covar(self, value: Tensor) -> None:
# internally uses a pivoted cholesky decomposition to construct a low rank
# approximation of the covariance
if self.rank > 0:
with torch.no_grad():
self.task_noise_covar_factor.data = to_linear_operator(value).pivoted_cholesky(rank=self.rank)
else:
raise AttributeError("Cannot set non-diagonal task noises when covariance is diagonal.")
def _eval_covar_matrix(self) -> Tensor:
covar_factor = self.task_noise_covar_factor
noise = self.noise
D = noise * torch.eye(self.num_tasks, dtype=noise.dtype, device=noise.device) # pyre-fixme[16]
return covar_factor.matmul(covar_factor.transpose(-1, -2)) + D
[docs] def marginal(
self, function_dist: MultitaskMultivariateNormal, *args: Any, **kwargs: Any
) -> MultitaskMultivariateNormal:
r"""
:return: Analytic marginal :math:`p(\mathbf y)`.
"""
return super().marginal(function_dist, *args, **kwargs)