#!/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.
r"""
Preference Learning with Gaussian Process
.. [Chu2005preference]
Wei Chu, and Zoubin Ghahramani. Preference learning with Gaussian processes.
Proceedings of the 22nd international conference on Machine learning. 2005.
.. [Brochu2010tutorial]
Eric Brochu, Vlad M. Cora, and Nando De Freitas.
A tutorial on Bayesian optimization of expensive cost functions,
with application to active user modeling and hierarchical reinforcement learning.
arXiv preprint arXiv:1012.2599 (2010).
"""
from __future__ import annotations
import math
import warnings
from copy import deepcopy
from typing import Any, List, Optional, Tuple, Union
import numpy as np
import torch
from botorch.models.model import Model
from botorch.posteriors.gpytorch import GPyTorchPosterior
from botorch.posteriors.posterior import Posterior
from gpytorch import settings
from gpytorch.constraints import GreaterThan, Positive
from gpytorch.distributions.multivariate_normal import MultivariateNormal
from gpytorch.kernels.rbf_kernel import RBFKernel
from gpytorch.lazy.added_diag_lazy_tensor import AddedDiagLazyTensor
from gpytorch.lazy.lazy_tensor import LazyTensor
from gpytorch.likelihoods.noise_models import HomoskedasticNoise
from gpytorch.means.constant_mean import ConstantMean
from gpytorch.mlls import MarginalLogLikelihood
from gpytorch.models.gp import GP
from gpytorch.module import Module
from gpytorch.priors.smoothed_box_prior import SmoothedBoxPrior
from gpytorch.priors.torch_priors import GammaPrior
from gpytorch.utils.cholesky import psd_safe_cholesky
from scipy import optimize
from torch import Tensor, float32, float64
[docs]class PairwiseGP(Model, GP):
r""" Probit GP for preference learning with Laplace approximation
Implementation is based on [Chu2005preference]_.
Also see [Brochu2010tutorial]_ for additional reference.
"""
def __init__(
self,
datapoints: Tensor,
comparisons: Tensor,
covar_module: Optional[Module] = None,
**kwargs,
) -> None:
super().__init__()
r"""A probit-likelihood GP with Laplace approximation model.
A probit-likelihood GP with Laplace approximation model that learns via
pairwise comparison data. By default it uses a scaled-RBF kernel.
Args:
datapoints: A `batch_shape x n x d` tensor of training features.
comparisons: A `batch_shape x m x 2` training comparisons;
comparisons[i] is a noisy indicator suggesting the utility value
of comparisons[i, 0]-th is greater than comparisons[i, 1]-th.
covar_module: Covariance module
"""
# Compatibility variables with fit_gpytorch_*: Dummy likelihood
# Likelihood is tightly tied with this model and
# it doesn't make much sense to keep it separate
self.likelihood = None
self.datapoints = None
self.comparisons = None
self.train_inputs = []
self.train_targets = None
self.pred_cov_fac_need_update = True
self._input_batch_shape = torch.Size()
self.dim = None
self.tkwargs = {}
# See set_train_data for additional compatibility variables
self.set_train_data(datapoints, comparisons, update_model=False)
self.covar = None
self.covar_inv = None
self.covar_chol = None
self.likelihood_hess = None
self.utility = None
self.hlcov_eye = None
self._std_norm = torch.distributions.normal.Normal(
torch.zeros(1, **self.tkwargs), torch.ones(1, **self.tkwargs)
)
# Set optional parameters
# jitter to add for numerical stability
self._jitter = kwargs.get("jitter", 1e-6)
# Clamping z lim for better numerical stability. See self._calc_z for detail
# norm_cdf(z=3) ~= 0.999, top 0.1% percent
self._zlim = kwargs.get("zlim", 3)
# Stopping creteria in scipy.optimize.fsolve used to find f_map in _update()
# If None, set to 1e-6 by default in _update
self._xtol = kwargs.get("xtol")
# The maximum number of calls to the function in scipy.optimize.fsolve
# If None, set to 100 by default in _update
# If zero, then 100*(N+1) is used by default by fsolve;
self._maxfev = kwargs.get("maxfev")
# Set hyperparameters
# Do not set the batch_shape explicitly so mean_module can operate in both mode
# once fsolve used in _update can run in batch mode, we should explicitly set
# the bacth shape here
self.mean_module = ConstantMean()
# Do not optimize constant mean prior
for param in self.mean_module.parameters():
param.requires_grad = False
self.noise_covar = HomoskedasticNoise(
noise_prior=SmoothedBoxPrior(-5, 5, 0.5, transform=torch.log),
noise_constraint=GreaterThan(1e-4), # if None, 1e-4 by default
batch_shape=self._input_batch_shape,
)
if covar_module is None:
ls_prior = GammaPrior(1.2, 0.5)
ls_prior_mode = (ls_prior.concentration - 1) / ls_prior.rate
self.covar_module = RBFKernel(
batch_shape=self._input_batch_shape,
ard_num_dims=self.dim,
lengthscale_prior=ls_prior,
lengthscale_constraint=Positive(
transform=None, initial_value=ls_prior_mode
),
)
else:
self.covar_module = covar_module
self._x0 = None # will store temporary results for warm-starting
if self.datapoints is not None and self.comparisons is not None:
self._update() # Find f_map for initial parameters
self.to(self.datapoints)
def __deepcopy__(self, memo) -> PairwiseGP:
attrs = (
"datapoints",
"comparisons",
"covar",
"covar_inv",
"covar_chol",
"likelihood_hess",
"utility",
"hlcov_eye",
)
if any(getattr(self, attr) is not None for attr in attrs):
# Temporarily remove non-leaf tensors so that pytorch allows deepcopy
old_attr = {}
for attr in attrs:
old_attr[attr] = getattr(self, attr)
setattr(self, attr, None)
new_model = deepcopy(self, memo)
# now set things back
for attr in attrs:
setattr(self, attr, old_attr[attr])
return new_model
else:
dcp = self.__deepcopy__
# make sure we don't fall into the infinite recursive loop
self.__deepcopy__ = None
new_model = deepcopy(self, memo)
self.__deepcopy__ = dcp
return new_model
@property
def std_noise(self) -> Tensor:
return self.noise_covar.noise
@std_noise.setter
def std_noise(self, value: Tensor) -> None:
self.noise_covar.initialize(noise=value)
@property
def num_outputs(self) -> int:
r"""The number of outputs of the model."""
return self._num_outputs
def _has_no_data(self):
r"""Return true if the model does not have both datapoints and comparisons"""
return (
self.datapoints is None
or len(self.datapoints.size()) == 0
or self.comparisons is None
)
def _calc_covar(
self, X1: Tensor, X2: Tensor, observation_noise: bool = False
) -> Union[Tensor, LazyTensor]:
r"""Calculate the covariance matrix given two sets of datapoints"""
X1, X2 = X1, X2
covar = self.covar_module(X1, X2)
if observation_noise:
noise_shape = self._input_batch_shape + self.covar.shape[-1:]
noise = self.noise_covar(shape=noise_shape)
covar = AddedDiagLazyTensor(covar, noise)
return covar.evaluate()
def _batch_chol_inv(self, mat_chol: Tensor) -> Tensor:
r"""Wrapper to perform (batched) cholesky inverse"""
# TODO: get rid of this once cholesky_inverse supports batch mode
batch_eye = torch.eye(mat_chol.shape[-1], **self.tkwargs)
if len(mat_chol.shape) == 2:
mat_inv = torch.cholesky_inverse(mat_chol)
elif len(mat_chol.shape) > 2 and (mat_chol.shape[-1] == mat_chol.shape[-2]):
batch_eye = batch_eye.repeat(*(mat_chol.shape[:-2]), 1, 1)
chol_inv = torch.triangular_solve(batch_eye, mat_chol, upper=False).solution
mat_inv = chol_inv.transpose(-1, -2) @ chol_inv
return mat_inv
def _update_covar(self) -> None:
r"""Update values derived from the data and hyperparameters
covar, covar_chol, and covar_inv will be of shape batch_shape x n x n
"""
self.covar = self._calc_covar(self.datapoints, self.datapoints)
self.covar_chol = psd_safe_cholesky(self.covar)
self.covar_inv = self._batch_chol_inv(self.covar_chol)
def _prior_mean(self, X: Tensor) -> Union[Tensor, LazyTensor]:
r"""Return point prediction using prior only
Args:
X: A `batch_size x n' x d`-dim Tensor at which to evaluate prior
Returns:
Prior mean prediction
"""
return self.mean_module(X)
def _prior_predict(self, X: Tensor) -> Tuple[Tensor, Tensor]:
r"""Predict utility based on prior info only
Args:
X: A `batch_size x n' x d`-dim Tensor at which to evaluate prior
Returns:
pred_mean: predictive mean
pred_covar: predictive covariance
"""
pred_mean = self._prior_mean(X)
pred_covar = self._calc_covar(X, X)
return pred_mean, pred_covar
def _add_jitter(self, X: Tensor) -> Tensor:
jitter_prev = 0
Eye = torch.eye(X.size(-1)).expand(X.shape)
for i in range(3):
jitter_new = self._jitter * (10 ** i)
X = X + (jitter_new - jitter_prev) * Eye
jitter_prev = jitter_new
# This may be VERY slow given upstream pytorch issue:
# https://github.com/pytorch/pytorch/issues/34272
try:
_ = torch.cholesky(X)
warnings.warn(
"X is not a p.d. matrix; "
f"Added jitter of {jitter_new:.2e} to the diagonal",
RuntimeWarning,
)
return X
except RuntimeError:
continue
warnings.warn(
f"Failed to render X p.d. after adding {jitter_new:.2e} jitter",
RuntimeWarning,
)
return X
def _calc_z(
self, utility: Tensor, D: Tensor, std_noise: Tensor
) -> Tuple[Tensor, Tensor, Tensor, Tensor]:
r"""Calculate z score.
Calculate z score as in [Chu2005preference]_: the standarized difference
Args:
utility: A Tensor of shape `batch_size x n`, the utility at MAP point
D: as in self.D
std_noise: comparison noise (as a Tensor since it's a pytorch param)
Returns:
z: z score calculated as in [Chu2005preference]_.
z_logpdf: log PDF of z
z_logcdf: log CDF of z
hazard: hazard function defined as pdf(z)/cdf(z)
"""
scaled_util = (utility / (math.sqrt(2) * std_noise)).unsqueeze(-1)
z = (D @ scaled_util).squeeze(-1)
# Clamp z for stable log transformation. This also prevent extreme values
# from appearing in the hess matrix, which should help with numerical
# stability and avoid extreme fitted hyperparameters
z = z.clamp(-self._zlim, self._zlim)
z_logpdf = self._std_norm.log_prob(z)
z_cdf = self._std_norm.cdf(z)
z_logcdf = torch.log(z_cdf)
hazard = torch.exp(z_logpdf - z_logcdf)
return z, z_logpdf, z_logcdf, hazard
def _grad_likelihood_f_sum(
self, utility: Tensor, D: Tensor, std_noise: Tensor
) -> Tensor:
r"""Compute the sum over of grad. of negative Log-LH wrt utility f.
Original grad should be of dimension m x n, as in (6) from [Chu2005preference]_.
Sum over the first dimension and return a tensor of shape n
Needed for calculating utility value at f_map to fill in torch gradient
Args:
utility: A Tensor of shape `batch_size x n`
D: A Tensor of shape `batch_size x m x n` as in self.D
std_noise: A Tensor of shape `batch_size x 1`, as in self.std_noise
Returns:
The sum over the first dimension of grad. of negative Log-LH wrt utility f
"""
_, _, _, h = self._calc_z(utility, D, std_noise)
h_factor = (h / (math.sqrt(2) * std_noise)).unsqueeze(-2)
grad = (h_factor @ (-D)).squeeze(-2)
return grad
def _hess_likelihood_f_sum(
self, utility: Tensor, D: Tensor, DT: Tensor, std_noise: Tensor
) -> Tensor:
r"""Compute the sum over of hessian of neg. Log-LH wrt utility f.
Original hess should be of dimension m x n x n, as in (7) from
[Chu2005preference]_ Sum over the first dimension and return a tensor of
shape n x n.
Args:
utility: A Tensor of shape `batch_size x n`
D: A Tensor of shape `batch_size x m x n` as in self.D
DT: Transpose of D. A Tensor of shape `batch_size x n x m` as in self.DT
std_noise: A Tensor of shape `batch_size x 1`, as in self.std_noise
Returns:
The sum over the first dimension of hess. of negative Log-LH wrt utility f
"""
z, _, _, h = self._calc_z(utility, D, std_noise)
mul_factor = h * (h + z) / (2 * (std_noise ** 2))
weighted_DT = DT * mul_factor.unsqueeze(-2).expand(*DT.size())
hess = weighted_DT @ D
return hess
def _grad_posterior_f(
self,
utility: Union[Tensor, np.ndarray],
datapoints: Tensor,
D: Tensor,
DT: Tensor,
std_noise: Tensor,
covar_chol: Tensor,
covar_inv: Tensor,
ret_np: bool = False,
) -> Union[Tensor, np.ndarray]:
r""" Compute the gradient of S loss wrt to f/utility in [Chu2005preference]_.
For finding f_map, which is negative of the log posterior, i.e., -log(p(f|D))
Derivative of (10) in [Chu2005preference]_.
Also see [Brochu2010tutorial]_ page 26. This is needed for estimating f_map.
Args:
utility: A Tensor of shape `batch_size x n`
datapoints: A Tensor of shape `batch_size x n x d` as in self.datapoints
D: A Tensor of shape `batch_size x m x n` as in self.D
DT: Transpose of D. A Tensor of shape `batch_size x n x m` as in self.DT
std_noise: A Tensor of shape `batch_size x 1`, as in self.std_noise
covar_chol: A Tensor of shape `batch_size x n x n`, as in self.covar_chol
covar_inv: A Tensor of shape `batch_size x n x n`, as in self.covar_inv
ret_np: return a numpy array if true, otherwise a Tensor
"""
if ret_np:
utility = torch.tensor(utility, **self.tkwargs)
b = self._grad_likelihood_f_sum(utility, D, std_noise)
# g_ = covar_inv x (utility - pred_prior)
p = (utility - self._prior_mean(datapoints)).unsqueeze(-1)
g_ = torch.cholesky_solve(p, covar_chol).squeeze(-1)
g = g_ + b
if ret_np:
return g.numpy()
else:
return g
def _hess_posterior_f(
self,
utility: Union[Tensor, np.ndarray],
datapoints: Tensor,
D: Tensor,
DT: Tensor,
std_noise: Tensor,
covar_chol: Tensor,
covar_inv: Tensor,
ret_np: bool = False,
) -> Union[Tensor, np.ndarray]:
r"""Compute the hessian of S loss wrt utility for finding f_map.
which is negative of the log posterior, i.e., -log(p(f|D))
Following [Chu2005preference]_ section 2.2.1.
This is needed for estimating f_map
Args:
utility: A Tensor of shape `batch_size x n`
datapoints: A Tensor of shape `batch_size x n x d` as in self.datapoints
D: A Tensor of shape `batch_size x m x n` as in self.D
DT: Transpose of D. A Tensor of shape `batch_size x n x m` as in self.DT
std_noise: A Tensor of shape `batch_size x 1`, as in self.std_noise
covar_chol: A Tensor of shape `batch_size x n x n`, as in self.covar_chol
covar_inv: A Tensor of shape `batch_size x n x n`, as in self.covar_inv
ret_np: return a numpy array if true, otherwise a Tensor
"""
if ret_np:
utility = torch.tensor(utility, **self.tkwargs)
hl = self._hess_likelihood_f_sum(utility, D, DT, std_noise)
hess = hl + covar_inv
return hess.numpy() if ret_np else hess
def _posterior_f(self, utility: Union[Tensor, np.ndarray]) -> Tensor:
r"""Calculate the negative of the log posterior, i.e., -log(P(f|D)).
This is the S loss function as in equation (10) of [Chu2005preference]_.
Args:
utility: A Tensor of shape `batch_size x n`
"""
_, _, z_logcdf, _ = self._calc_z(utility, self.D, self.std_noise)
loss1 = -(torch.sum(z_logcdf, dim=-1))
inv_prod = torch.cholesky_solve(utility.unsqueeze(-1), self.covar_chol)
loss2 = 0.5 * (utility.unsqueeze(-2) @ inv_prod).squeeze(-1).squeeze(-1)
loss = loss1 + loss2
loss = loss.clamp(min=0)
return loss
def _update_utility_derived_values(self) -> None:
r"""Calculate utility-derived values not needed during optimization
Using subsitution method for better numerical stability
Let `pred_cov_fac = (covar + hl^-1)`, which is needed for calculate
predictive covariance = `K - k.T @ pred_cov_fac^-1 @ k`
(Also see posterior mode in `forward`)
Instead of inverting `pred_cov_fac`, let `hlcov_eye = (hl @ covar + I)`
Then we can obtain `pred_cov_fac^-1 @ k` by solving for p in
`(hl @ k) p = hlcov_eye`
`hlcov_eye p = hl @ k`
"""
hl = self.likelihood_hess # "C" from page 27, [Brochu2010tutorial]_
hlcov = hl @ self.covar
eye = torch.eye(hlcov.size(-1)).expand(hlcov.shape)
self.hlcov_eye = hlcov + eye
self.pred_cov_fac_need_update = False
def _update(self, **kwargs) -> None:
r"""Update the model by updating the covar matrix and MAP utility values
Update the model by
1. Re-evaluating the covar matrix as the data or hyperparams may have changed
2. Approximating maximum a posteriori of the utility function f using fsolve
Should be called after data or hyperparameters are changed to update
f_map and related values
self._xtol and self._maxfev are passed to fsolve as xtol and maxfev
to control stopping criteria
"""
xtol = 1e-6 if self._xtol is None else self._xtol
maxfev = 100 if self._maxfev is None else self._maxfev
# Using the latest param for covariance before calculating f_map
self._update_covar()
# scipy newton raphson
with torch.no_grad():
# warm start
init_x0_size = self._input_batch_shape + torch.Size([self.n])
if self._x0 is None or torch.Size(self._x0.shape) != init_x0_size:
x0 = np.random.rand(*init_x0_size)
else:
x0 = self._x0
if len(self._input_batch_shape) > 0:
# batch mode, do optimize.fsolve sequentially
# TODO: enable vectorization/parallelization here
x0 = x0.reshape(-1, self.n)
dp_v = self.datapoints.view(-1, self.n, self.dim)
D_v = self.D.view(-1, self.m, self.n)
DT_v = self.DT.view(-1, self.n, self.m)
# Use `expand` here since we need to expand std_noise along
# the batch shape dimensions if we start off as non-batch model,
# but later conditioned on batched new data
sn_v = self.std_noise.expand(*init_x0_size[:-1], 1).reshape(-1)
ch_v = self.covar_chol.view(-1, self.n, self.n)
ci_v = self.covar_inv.view(-1, self.n, self.n)
x = np.empty(x0.shape)
for i in range(x0.shape[0]):
fsolve_args = (
dp_v[i],
D_v[i],
DT_v[i],
sn_v[i],
ch_v[i],
ci_v[i],
True,
)
with warnings.catch_warnings():
warnings.filterwarnings("ignore", category=RuntimeWarning)
x[i] = optimize.fsolve(
x0=x0[i],
func=self._grad_posterior_f,
fprime=self._hess_posterior_f,
xtol=xtol,
maxfev=maxfev,
args=fsolve_args,
**kwargs,
)
x = x.reshape(*init_x0_size)
else:
fsolve_args = (
self.datapoints,
self.D,
self.DT,
self.std_noise,
self.covar_chol,
self.covar_inv,
True,
)
with warnings.catch_warnings():
warnings.filterwarnings("ignore", category=RuntimeWarning)
x = optimize.fsolve(
x0=x0,
func=self._grad_posterior_f,
fprime=self._hess_posterior_f,
xtol=xtol,
maxfev=maxfev,
args=fsolve_args,
**kwargs,
)
self._x0 = x.copy() # save for warm-starting
f = torch.tensor(x, **self.tkwargs)
# To perform hyperparameter optimization, this need to be recalculated
# when calling forward() in order to obtain correct gradients
# self.likelihood_hess is updated here is for the rare case where we
# do not want to call forward()
self.likelihood_hess = self._hess_likelihood_f_sum(
f, self.D, self.DT, self.std_noise
)
# Lazy update utility pred_cov_fac
self.pred_cov_fac_need_update = True
self.utility = f.clone().requires_grad_(True)
def _transform_batch_shape(self, X: Tensor, X_new: Tensor) -> Tuple[Tensor, Tensor]:
r"""Transform X and X_new into the same shape
Transform the batch shape of X to be compatible
with `X_new` to calculate the posterior.
If X has the same batch size as `X_new`, return it as is.
If one is in batch mode and the other one is not, convert both
into batch mode.
If both are in batch mode, this will only work if X_batch_shape
can propagate to X_new_batch_shape
Args:
X: A `batch_shape x q x d`-dim or `(1 x) q x d`-dim Tensor
X_new: A `batch_shape x q x d`-dim Tensor
Returns:
Transformed X and X_new pair
"""
X_bs = X.shape[:-2] # X batch shape
X_new_bs = X_new.shape[:-2] # X_new batch shape
if X_new_bs == X_bs:
# if batch shapes match, there's no need to transform
# X_new may or may not have batch_shape dimensions
return X, X_new
elif len(X_new_bs) < len(X_bs):
# if X_new has fewer dimension, try to expand it to X's shape
return X, X_new.expand(X_bs + X_new.shape[-2:])
else:
# if X has fewer dimension, try to expand it to X_new's shape
return X.expand(X_new_bs + X.shape[-2:]), X_new
def _util_newton_updates(self, x0, max_iter=100, xtol=None) -> Tensor:
r"""Make `max_iter` newton updates on utility.
This is used in `forward` to calculate and fill in gradient into tensors.
Instead of doing utility -= H^-1 @ g, use substition method.
See more explanation in _update_utility_derived_values.
Args:
x0: A `batch_size x n` dimension tensor, initial values
max_iter: Max number of iterations
xtol: Stop creteria. If `None`, do not stop until
finishing `max_iter` updates
"""
xtol = float("-Inf") if xtol is None else xtol
dp, D, DT, sn, ch, ci = (
self.datapoints,
self.D,
self.DT,
self.std_noise,
self.covar_chol,
self.covar_inv,
)
covar = self.covar
diff = float("Inf")
i = 0
x = x0
eye = None
while i < max_iter and diff > xtol:
hl = self._hess_likelihood_f_sum(x, D, DT, sn)
cov_hl = covar @ hl
if eye is None:
eye = torch.eye(cov_hl.size(-1)).expand(cov_hl.shape)
cov_hl = cov_hl + eye # add 1 to cov_hl
g = self._grad_posterior_f(x, dp, D, DT, sn, ch, ci)
cov_g = covar @ g.unsqueeze(-1)
x_update = torch.solve(cov_g, cov_hl).solution.squeeze(-1)
x_next = x - x_update
diff = torch.norm(x - x_next)
x = x_next
i += 1
return x
# ============== public APIs ==============
[docs] def set_train_data(
self, datapoints: Tensor, comparisons: Tensor, update_model: bool = True
) -> None:
r"""Set datapoints and comparisons and update model properties if needed
Args:
datapoints: A `batch_shape x n x d` dimension tensor X
comparisons: A tensor of size `batch_shape x m x 2`. (i, j) means
f_i is preferred over f_j
update_model: True if we want to refit the model (see _update) after
re-setting the data
"""
# When datapoints and/or comparisons are None, we are constructing
# a prior-only model
if datapoints is None or comparisons is None:
return
# store datapoints and other quantities as double for better numerical stability
self.datapoints = datapoints
# convert to long so that it can be used as index and work with Tensor.scatter_
self.comparisons = comparisons.long()
self.tkwargs = {
"dtype": self.datapoints.dtype,
"device": self.datapoints.device,
}
self._std_norm = torch.distributions.normal.Normal(
torch.zeros(1, **self.tkwargs), torch.ones(1, **self.tkwargs)
)
# Compatibility variables with fit_gpytorch_*
# alias for datapoints (train_inputs) and comparisons ("train_targets" here)
self.train_inputs = [self.datapoints]
self.train_targets = self.comparisons
# Compatibility variables with optimize_acqf
self._dtype = self.datapoints.dtype
self._num_outputs = 1 # 1 latent value output per observation
self._input_batch_shape = self.datapoints.shape[:-2]
self.dim = self.datapoints.shape[-1] # feature dimensions
self.n = self.datapoints.shape[-2] # num datapoints
self.m = self.comparisons.shape[-2] # num pairwise comparisons
self.utility = None
# D is batch_sizem x n or num_comparison x num_datapoints.
# D_k_i is the s_k(x_i) value as in equation (6) in [Chu2005preference]_
# D will usually be very sparse as well
# TODO swap out scatter_ so that comparisons could be int instead of long
# TODO: make D a sparse matrix once pytorch has better support for
# sparse tensors
D_size = torch.Size((*(self._input_batch_shape), self.m, self.n))
self.D = torch.zeros(D_size, **self.tkwargs)
comp_view = self.comparisons.view(-1, self.m, 2).long()
for i, sub_D in enumerate(self.D.view(-1, self.m, self.n)):
sub_D.scatter_(1, comp_view[i, :, [0]], 1)
sub_D.scatter_(1, comp_view[i, :, [1]], -1)
self.DT = self.D.transpose(-1, -2)
if update_model:
self._update()
self.to(self.datapoints)
[docs] def forward(self, datapoints: Tensor) -> MultivariateNormal:
r"""Calculate a posterior or prior prediction.
During training mode, forward implemented solely for gradient-based
hyperparam opt. Essentially what it does is to re-calculate the utility
f using its analytical form at f_map so that we are able to obtain
gradients of the hyperparameters.
We only take in one parameter datapoints without the comparisons for the
compatibility with other gpytorch/botorch APIs. It assumes `datapoints` is
the same as `self.datapoints`. That's what "Must train on training data" means.
Args:
datapoints: A `batch_shape x n x d` Tensor,
should be the same as self.datapoints
Returns:
A MultivariateNormal object, being one of the followings:
1. Posterior centered at MAP points for training data (training mode)
2. Prior predictions (prior mode)
3. Predictive posterior (eval mode)
"""
# Training mode: optimizing
if self.training:
if self._has_no_data():
raise RuntimeError(
"datapoints and comparisons cannot be None in training mode. "
"Call .eval() for prior predictions, "
"or call .set_train_data() to add training data."
)
if datapoints is not self.datapoints:
raise RuntimeError("Must train on training data")
self.set_train_data(datapoints, self.comparisons, update_model=True)
# Take a newton step on the posterior MAP point to fill
# in gradients for pytorch
self.utility = self._util_newton_updates(self.utility, max_iter=1)
hl = self.likelihood_hess = self._hess_likelihood_f_sum(
self.utility, self.D, self.DT, self.std_noise
)
covar = self.covar
# Apply matrix inversion lemma on eq. in page 27 of [Brochu2010tutorial]_
# (A + B)^-1 = A^-1 - A^-1 @ (I + BA^-1)^-1 @ BA^-1
# where A = covar_inv, B = hl
hl_cov = hl @ covar
eye = torch.eye(hl_cov.size(-1)).expand(hl_cov.shape)
hl_cov_I = hl_cov + eye # add I to hl_cov
train_covar_map = covar - covar @ torch.solve(hl_cov, hl_cov_I).solution
output_mean, output_covar = self.utility, train_covar_map
# Prior mode
elif settings.prior_mode.on() or self._has_no_data():
X_new = datapoints
# if we don't have any data yet, use prior GP to make predictions
output_mean, output_covar = self._prior_predict(X_new)
# Posterior mode
else:
# self.utility might be None if exception was raised and _update
# was failed to be called during hyperparameter optimization
# procedures (e.g., fit_gpytorch_scipy)
if self.utility is None:
self._update()
if self.pred_cov_fac_need_update:
self._update_utility_derived_values()
datapoints = datapoints.to(self.datapoints)
X, X_new = self._transform_batch_shape(self.datapoints, datapoints)
covar_chol, _ = self._transform_batch_shape(self.covar_chol, X_new)
hl, _ = self._transform_batch_shape(self.likelihood_hess, X_new)
hlcov_eye, _ = self._transform_batch_shape(self.hlcov_eye, X_new)
# otherwise compute predictive mean and covariance
covar_xnew_x = self._calc_covar(X_new, X)
covar_x_xnew = covar_xnew_x.transpose(-1, -2)
covar_xnew = self._calc_covar(X_new, X_new)
p = self.utility - self._prior_mean(X)
covar_inv_p = torch.cholesky_solve(p.unsqueeze(-1), covar_chol)
pred_mean = (covar_xnew_x @ covar_inv_p).squeeze(-1)
pred_mean = pred_mean + self._prior_mean(X_new)
# [Brochu2010tutorial]_ page 27
# Preictive covariance fatcor: hlcov_eye = (K + C^-1)
# fac = (K + C^-1)^-1 @ k = pred_cov_fac_inv @ covar_x_xnew
# used substitution method here to calculate fac
fac = torch.solve(hl @ covar_x_xnew, hlcov_eye).solution
pred_covar = covar_xnew - (covar_xnew_x @ fac)
output_mean, output_covar = pred_mean, pred_covar
try:
diag_jitter = torch.eye(output_covar.size(-1)).expand(output_covar.shape)
diag_jitter = diag_jitter * self._jitter
# Preemptively adding jitter to diagonal to prevent the use of _add_jitter
# given that torch.cholesky may be very slow on non-pd matrix input
# See https://github.com/pytorch/pytorch/issues/34272
# TODO: remove this once torch.cholesky issue is resolved
output_covar = output_covar + diag_jitter
post = MultivariateNormal(output_mean, output_covar)
except RuntimeError:
output_covar = self._add_jitter(output_covar)
post = MultivariateNormal(output_mean, output_covar)
return post
# ============== botorch.models.model.Model interfaces ==============
[docs] def posterior(
self,
X: Tensor,
output_indices: Optional[List[int]] = None,
observation_noise: bool = False,
**kwargs: Any,
) -> Posterior:
r"""Computes the posterior over model outputs at the provided points.
Args:
X: A `batch_shape x q x d`-dim Tensor, where `d` is the dimension
of the feature space and `q` is the number of points considered jointly.
output_indices: As defined in parent Model class, not used for this model.
observation_noise: If True, add observation noise to the posterior.
Returns:
A `Posterior` object, representing joint
distributions over `q` points. Includes observation noise if specified.
"""
self.eval() # make sure model is in eval mode
if output_indices is not None:
raise RuntimeError(
"output_indices is not None. PairwiseGP should not be a"
"multi-output model."
)
post = self(X)
if observation_noise:
noise_covar = self.noise_covar(shape=post.mean.shape).evaluate()
post = MultivariateNormal(post.mean, post.covariance_matrix + noise_covar)
return GPyTorchPosterior(post)
[docs] def condition_on_observations(self, X: Tensor, Y: Tensor, **kwargs: Any) -> Model:
r"""Condition the model on new observations.
Note that unlike other BoTorch models, PairwiseGP requires Y to be
pairwise comparisons
Args:
X: A `batch_shape x n x d` dimension tensor X
Y: A tensor of size `batch_shape x m x 2`. (i, j) means
f_i is preferred over f_j
Returns:
A (deepcopied) `Model` object of the same type, representing the
original model conditioned on the new observations `(X, Y)` (and
possibly noise observations passed in via kwargs).
"""
new_model = deepcopy(self)
if self._has_no_data():
# If the model previously has no data, set X and Y as the data directly
new_model.set_train_data(X, Y, update_model=True)
else:
# Can only condition on pairwise comparisons instead of the directly
# observed values. Raise a RuntimeError if Y is not a tensor presenting
# pairwise comparisons
if Y.dtype in (float32, float64) or Y.shape[-1] != 2:
raise RuntimeError(
"Conditioning on non-pairwise comparison observations."
)
# Reshaping datapoints and comparisons by batches
Y_new_batch_shape = Y.shape[:-2]
new_datapoints = self.datapoints.expand(
Y_new_batch_shape + self.datapoints.shape[-2:]
)
new_comparisons = self.comparisons.expand(
Y_new_batch_shape + self.comparisons.shape[-2:]
)
# Reshape X since Y may have additional batch dim. from fantasy models
X = X.expand(Y_new_batch_shape + X.shape[-2:])
new_datapoints = torch.cat((new_datapoints, X.to(new_datapoints)), dim=-2)
shifted_comp = Y.to(new_comparisons) + self.n
new_comparisons = torch.cat((new_comparisons, shifted_comp), dim=-2)
# TODO: be smart about how we can update covar matrix here
new_model.set_train_data(new_datapoints, new_comparisons, update_model=True)
return new_model
[docs]class PairwiseLaplaceMarginalLogLikelihood(MarginalLogLikelihood):
def __init__(self, model: PairwiseGP) -> None:
r"""Laplace-approximated marginal log likelihood/evidence for PairwiseGP
See (12) from [Chu2005preference]_.
Args:
model: A model using laplace approximation (currently only supports
`PairwiseGP`)
"""
# Do not use likelihood module here as it's implicitly included in the model
super().__init__(None, model)
[docs] def forward(self, post: Posterior, comp: Tensor) -> Tensor:
r"""Calculate approximated log evidence, i.e., log(P(D|theta))
Args:
post: training posterior distribution from self.model
comp: Comparisons pairs, see PairwiseGP.__init__ for more details
Returns:
The approximated evidence, i.e., the marginal log likelihood
"""
model = self.model
if comp is not model.comparisons:
raise RuntimeError("Must train on training data")
f_max = post.mean
log_posterior = model._posterior_f(f_max)
part1 = -log_posterior
part2 = model.covar @ model.likelihood_hess
eye = torch.eye(part2.size(-1)).expand(part2.shape)
part2 = part2 + eye
part2 = -0.5 * torch.logdet(part2)
evidence = part1 + part2
# Sum up mll first so that when adding prior probs it won't
# propagate and double count
evidence = evidence.sum()
# Add log probs of priors on the (functions of) parameters
for _, prior, closure, _ in self.named_priors():
evidence = evidence.add(prior.log_prob(closure()).sum())
return evidence
