#!/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"""
Analytic Acquisition Functions for Multi-objective Bayesian optimization.
References
.. [Yang2019]
Yang, K., Emmerich, M., Deutz, A. et al. Efficient computation of expected
hypervolume improvement using box decomposition algorithms. J Glob Optim 75,
3–34 (2019)
"""
from __future__ import annotations
from abc import abstractmethod
from itertools import product
from typing import List, Optional
import torch
from botorch.acquisition.acquisition import AcquisitionFunction
from botorch.acquisition.multi_objective.objective import (
AnalyticMultiOutputObjective,
IdentityAnalyticMultiOutputObjective,
)
from botorch.exceptions.errors import UnsupportedError
from botorch.models.model import Model
from botorch.utils.multi_objective.box_decomposition import NondominatedPartitioning
from botorch.utils.transforms import t_batch_mode_transform
from torch import Tensor
from torch.distributions import Normal
[docs]class MultiObjectiveAnalyticAcquisitionFunction(AcquisitionFunction):
r"""Abstract base class for Multi-Objective batch acquisition functions."""
def __init__(
self, model: Model, objective: Optional[AnalyticMultiOutputObjective] = None
) -> None:
r"""Constructor for the MultiObjectiveAnalyticAcquisitionFunction base class.
Args:
model: A fitted model.
objective: An AnalyticMultiOutputObjective (optional).
"""
super().__init__(model=model)
if objective is None:
objective = IdentityAnalyticMultiOutputObjective()
elif not isinstance(objective, AnalyticMultiOutputObjective):
raise UnsupportedError(
"Only objectives of type AnalyticMultiOutputObjective are supported "
"for Multi-Objective analytic acquisition functions."
)
self.objective = objective
[docs] @abstractmethod
def forward(self, X: Tensor) -> Tensor:
r"""Takes in a `batch_shape x 1 x d` X Tensor of t-batches with `1` `d`-dim
design point each, and returns a Tensor with shape `batch_shape'`, where
`batch_shape'` is the broadcasted batch shape of model and input `X`.
"""
pass # pragma: no cover
[docs] def set_X_pending(self, X_pending: Optional[Tensor] = None) -> None:
raise UnsupportedError(
"Analytic acquisition functions do not account for X_pending yet."
)
[docs]class ExpectedHypervolumeImprovement(MultiObjectiveAnalyticAcquisitionFunction):
def __init__(
self,
model: Model,
ref_point: List[float],
partitioning: NondominatedPartitioning,
objective: Optional[AnalyticMultiOutputObjective] = None,
) -> None:
r"""Expected Hypervolume Improvement supporting m>=2 outcomes.
This implements the computes EHVI using the algorithm from [Yang2019]_, but
additionally computes gradients via auto-differentiation as proposed by
[Daulton2020qehvi]_.
Note: this is currently inefficient in two ways due to the binary partitioning
algorithm that we use for the box decomposition:
- We have more boxes in our decomposition
- If we used a box decomposition that used `inf` as the upper bound for
the last dimension *in all hypercells*, then we could reduce the number
of terms we need to compute from 2^m to 2^(m-1). [Yang2019]_ do this
by using DKLV17 and LKF17 for the box decomposition.
TODO: Use DKLV17 and LKF17 for the box decomposition as in [Yang2019]_ for
greater efficiency.
TODO: Add support for outcome constraints.
Example:
>>> model = SingleTaskGP(train_X, train_Y)
>>> ref_point = [0.0, 0.0]
>>> EHVI = ExpectedHypervolumeImprovement(model, ref_point, partitioning)
>>> ehvi = EHVI(test_X)
Args:
model: A fitted model.
ref_point: A list with `m` elements representing the reference point (in the
outcome space) w.r.t. to which compute the hypervolume. This is a
reference point for the objective values (i.e. after applying
`objective` to the samples).
partitioning: A `NondominatedPartitioning` module that provides the non-
dominated front and a partitioning of the non-dominated space in hyper-
rectangles.
objective: An `AnalyticMultiOutputObjective`.
"""
# TODO: we could refactor this __init__ logic into a
# HypervolumeAcquisitionFunction Mixin
if len(ref_point) != partitioning.num_outcomes:
raise ValueError(
"The length of the reference point must match the number of outcomes. "
f"Got ref_point with {len(ref_point)} elements, but expected "
f"{partitioning.num_outcomes}."
)
ref_point = torch.tensor(
ref_point,
dtype=partitioning.pareto_Y.dtype,
device=partitioning.pareto_Y.device,
)
better_than_ref = (partitioning.pareto_Y > ref_point).all(dim=1)
if not better_than_ref.any() and partitioning.pareto_Y.shape[0] > 0:
raise ValueError(
"At least one pareto point must be better than the reference point."
)
super().__init__(model=model, objective=objective)
self.register_buffer("ref_point", ref_point)
self.partitioning = partitioning
cell_bounds = self.partitioning.get_hypercell_bounds(ref_point=self.ref_point)
self.register_buffer("cell_lower_bounds", cell_bounds[0])
self.register_buffer("cell_upper_bounds", cell_bounds[1])
# create indexing tensor of shape `2^m x m`
self._cross_product_indices = torch.tensor(
list(product(*[[0, 1] for _ in range(ref_point.shape[0])])),
dtype=torch.long,
device=ref_point.device,
)
self.normal = Normal(0, 1)
[docs] def psi(self, lower: Tensor, upper: Tensor, mu: Tensor, sigma: Tensor) -> None:
r"""Compute Psi function.
For each cell i and outcome k:
Psi(lower_{i,k}, upper_{i,k}, mu_k, sigma_k) = (
sigma_k * PDF((upper_{i,k} - mu_k) / sigma_k) + (
mu_k - lower_{i,k}
) * (1 - CDF(upper_{i,k} - mu_k) / sigma_k)
)
See Equation 19 in [Yang2019]_ for more details.
Args:
lower: A `num_cells x m`-dim tensor of lower cell bounds
upper: A `num_cells x m`-dim tensor of upper cell bounds
mu: A `batch_shape x 1 x m`-dim tensor of means
sigma: A `batch_shape x 1 x m`-dim tensor of standard deviations (clamped).
Returns:
A `batch_shape x num_cells x m`-dim tensor of values.
"""
u = (upper - mu) / sigma
return sigma * self.normal.log_prob(u).exp() + (mu - lower) * (
1 - self.normal.cdf(u)
)
[docs] def nu(self, lower: Tensor, upper: Tensor, mu: Tensor, sigma: Tensor) -> None:
r"""Compute Nu function.
For each cell i and outcome k:
nu(lower_{i,k}, upper_{i,k}, mu_k, sigma_k) = (
upper_{i,k} - lower_{i,k}
) * (1 - CDF((upper_{i,k} - mu_k) / sigma_k))
See Equation 25 in [Yang2019]_ for more details.
Args:
lower: A `num_cells x m`-dim tensor of lower cell bounds
upper: A `num_cells x m`-dim tensor of upper cell bounds
mu: A `batch_shape x 1 x m`-dim tensor of means
sigma: A `batch_shape x 1 x m`-dim tensor of standard deviations (clamped).
Returns:
A `batch_shape x num_cells x m`-dim tensor of values.
"""
return (upper - lower) * (1 - self.normal.cdf((upper - mu) / sigma))
[docs] @t_batch_mode_transform()
def forward(self, X: Tensor) -> Tensor:
posterior = self.objective(self.model.posterior(X))
mu = posterior.mean
sigma = posterior.variance.clamp_min(1e-9).sqrt()
# clamp here, since upper_bounds will contain `inf`s, which
# are not differentiable
cell_upper_bounds = self.cell_upper_bounds.clamp_max(
1e10 if X.dtype == torch.double else 1e8
)
# Compute psi(lower_i, upper_i, mu_i, sigma_i) for i=0, ... m-2
psi_lu = self.psi(
lower=self.cell_lower_bounds, upper=cell_upper_bounds, mu=mu, sigma=sigma
)
# Compute psi(lower_m, lower_m, mu_m, sigma_m)
psi_ll = self.psi(
lower=self.cell_lower_bounds,
upper=self.cell_lower_bounds,
mu=mu,
sigma=sigma,
)
# Compute nu(lower_m, upper_m, mu_m, sigma_m)
nu = self.nu(
lower=self.cell_lower_bounds, upper=cell_upper_bounds, mu=mu, sigma=sigma
)
# compute the difference psi_ll - psi_lu
psi_diff = psi_ll - psi_lu
# this is batch_shape x num_cells x 2 x (m-1)
stacked_factors = torch.stack([psi_diff, nu], dim=-2)
# Take the cross product of psi_diff and nu across all outcomes
# e.g. for m = 2
# for each batch and cell, compute
# [psi_diff_0, psi_diff_1]
# [nu_0, psi_diff_1]
# [psi_diff_0, nu_1]
# [nu_0, nu_1]
# this tensor has shape: `batch_shape x num_cells x 2^m x m`
all_factors_up_to_last = stacked_factors.gather(
dim=-2,
index=self._cross_product_indices.expand(
stacked_factors.shape[:-2] + self._cross_product_indices.shape
),
)
# compute product for all 2^m terms,
# sum across all terms and hypercells
return all_factors_up_to_last.prod(dim=-1).sum(dim=-1).sum(dim=-1)
