#!/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"""Algorithms for partitioning the non-dominated space into rectangles.
References
.. [Couckuyt2012]
I. Couckuyt, D. Deschrijver and T. Dhaene, "Towards Efficient
Multiobjective Optimization: Multiobjective statistical criterions,"
2012 IEEE Congress on Evolutionary Computation, Brisbane, QLD, 2012,
pp. 1-8.
"""
from __future__ import annotations
from typing import Optional
import torch
from botorch.utils.multi_objective.box_decompositions.box_decomposition import (
BoxDecomposition,
FastPartitioning,
)
from botorch.utils.multi_objective.box_decompositions.utils import (
_expand_ref_point,
get_partition_bounds,
update_local_upper_bounds_incremental,
compute_non_dominated_hypercell_bounds_2d,
)
from torch import Tensor
[docs]class NondominatedPartitioning(BoxDecomposition):
r"""A class for partitioning the non-dominated space into hyper-cells.
Note: this assumes maximization. Internally, it multiplies outcomes by -1 and
performs the decomposition under minimization. TODO: use maximization
internally as well.
Note: it is only feasible to use this algorithm to compute an exact
decomposition of the non-dominated space for `m<5` objectives (alpha=0.0).
The alpha parameter can be increased to obtain an approximate partitioning
faster. The `alpha` is a fraction of the total hypervolume encapsuling the
entire Pareto set. When a hypercell's volume divided by the total hypervolume
is less than `alpha`, we discard the hypercell. See Figure 2 in
[Couckuyt2012]_ for a visual representation.
This PyTorch implementation of the binary partitioning algorithm ([Couckuyt2012]_)
is adapted from numpy/tensorflow implementation at:
https://github.com/GPflow/GPflowOpt/blob/master/gpflowopt/pareto.py.
TODO: replace this with a more efficient decomposition. E.g.
https://link.springer.com/content/pdf/10.1007/s10898-019-00798-7.pdf
"""
def __init__(
self,
ref_point: Tensor,
Y: Optional[Tensor] = None,
alpha: float = 0.0,
) -> None:
"""Initialize NondominatedPartitioning.
Args:
ref_point: A `m`-dim tensor containing the reference point.
Y: A `(batch_shape) x n x m`-dim tensor.
alpha: A thresold fraction of total volume used in an approximate
decomposition.
Example:
>>> bd = NondominatedPartitioning(ref_point, Y=Y1)
"""
self.alpha = alpha
super().__init__(ref_point=ref_point, sort=True, Y=Y)
def _partition_space(self) -> None:
r"""Partition the non-dominated space into disjoint hypercells.
This method supports an arbitrary number of outcomes, but is
less efficient than `partition_space_2d` for the 2-outcome case.
"""
# The binary parititoning algorithm uses indices the augmented Pareto front.
# n_pareto + 2 x m
aug_pareto_Y_idcs = self._get_augmented_pareto_front_indices()
# Initialize one cell over entire pareto front
cell = torch.zeros(
2, self.num_outcomes, dtype=torch.long, device=self._neg_Y.device
)
cell[1] = aug_pareto_Y_idcs.shape[0] - 1
stack = [cell]
# hypercells contains the indices of the (augmented) Pareto front
# that specify that bounds of the each hypercell.
# It is a `2 x num_cells x m`-dim tensor
self.register_buffer(
"hypercells",
torch.empty(
2, 0, self.num_outcomes, dtype=torch.long, device=self._neg_Y.device
),
)
outcome_idxr = torch.arange(
self.num_outcomes, dtype=torch.long, device=self._neg_Y.device
)
# edge case: empty pareto set
# use a single cell
if self._neg_pareto_Y.shape[-2] == 0:
# 2 x m
cell_bounds_pareto_idcs = aug_pareto_Y_idcs[cell, outcome_idxr]
self.hypercells = torch.cat(
[self.hypercells, cell_bounds_pareto_idcs.unsqueeze(1)], dim=1
)
else:
# Extend Pareto front with the ideal and anti-ideal point
ideal_point = self._neg_pareto_Y.min(dim=0, keepdim=True).values - 1
anti_ideal_point = self._neg_pareto_Y.max(dim=0, keepdim=True).values + 1
# `n_pareto + 2 x m`
aug_pareto_Y = torch.cat(
[ideal_point, self._neg_pareto_Y, anti_ideal_point], dim=0
)
total_volume = (anti_ideal_point - ideal_point).prod()
# Use binary partitioning
while len(stack) > 0:
# The following 3 tensors are all `2 x m`
cell = stack.pop()
cell_bounds_pareto_idcs = aug_pareto_Y_idcs[cell, outcome_idxr]
cell_bounds_pareto_values = aug_pareto_Y[
cell_bounds_pareto_idcs, outcome_idxr
]
# Check cell bounds
# - if cell upper bound is better than Pareto front on all outcomes:
# - accept the cell
# - elif cell lower bound is better than Pareto front on all outcomes:
# - this means the cell overlaps the Pareto front. Divide the cell
# along its longest edge.
if (
(cell_bounds_pareto_values[1] <= self._neg_pareto_Y)
.any(dim=1)
.all()
):
# Cell is entirely non-dominated
self.hypercells = torch.cat(
[self.hypercells, cell_bounds_pareto_idcs.unsqueeze(1)], dim=1
)
elif (
(cell_bounds_pareto_values[0] <= self._neg_pareto_Y)
.any(dim=1)
.all()
):
# The cell overlaps the pareto front
# compute the distance (in integer indices)
# This has shape `m`
idx_dist = cell[1] - cell[0]
any_not_adjacent = (idx_dist > 1).any()
cell_volume = (
(cell_bounds_pareto_values[1] - cell_bounds_pareto_values[0])
.prod(dim=-1)
.item()
)
# Only divide a cell when it is not composed of adjacent indices
# and the fraction of total volume is above the approximation
# threshold fraction
if (
any_not_adjacent
and ((cell_volume / total_volume) > self.alpha).all()
):
# Divide the test cell over its largest dimension
# largest (by index length)
length, longest_dim = torch.max(idx_dist, dim=0)
length = length.item()
longest_dim = longest_dim.item()
new_length1 = int(round(length / 2.0))
new_length2 = length - new_length1
# Store divided cells
# cell 1: subtract new_length1 from the upper bound of the cell
# cell 2: add new_length2 to the lower bound of the cell
for bound_idx, length_delta in (
(1, -new_length1),
(0, new_length2),
):
new_cell = cell.clone()
new_cell[bound_idx, longest_dim] += length_delta
stack.append(new_cell)
def _partition_space_2d(self) -> None:
r"""Partition the non-dominated space into disjoint hypercells.
This direct method works for `m=2` outcomes.
"""
pf_ext_idx = self._get_augmented_pareto_front_indices()
n_pf_plus_1 = self._neg_pareto_Y.shape[-2] + 1
view_shape = torch.Size([1] * len(self.batch_shape) + [n_pf_plus_1])
expand_shape = self.batch_shape + torch.Size([n_pf_plus_1])
range_pf_plus1 = torch.arange(
n_pf_plus_1, dtype=torch.long, device=self._neg_pareto_Y.device
)
range_pf_plus1_expanded = range_pf_plus1.view(view_shape).expand(expand_shape)
lower = torch.stack(
[range_pf_plus1_expanded, torch.zeros_like(range_pf_plus1_expanded)], dim=-1
)
upper = torch.stack(
[1 + range_pf_plus1_expanded, pf_ext_idx[..., -range_pf_plus1 - 1, -1]],
dim=-1,
)
# 2 x batch_shape x n_cells x 2
self.register_buffer("hypercells", torch.stack([lower, upper], dim=0))
def _get_augmented_pareto_front_indices(self) -> Tensor:
r"""Get indices of augmented Pareto front."""
pf_idx = torch.argsort(self._neg_pareto_Y, dim=-2)
return torch.cat(
[
torch.zeros(
*self.batch_shape,
1,
self.num_outcomes,
dtype=torch.long,
device=self._neg_Y.device,
),
# Add 1 because index zero is used for the ideal point
pf_idx + 1,
torch.full(
torch.Size(
[
*self.batch_shape,
1,
self.num_outcomes,
]
),
self._neg_pareto_Y.shape[-2] + 1,
dtype=torch.long,
device=self._neg_Y.device,
),
],
dim=-2,
)
[docs] def get_hypercell_bounds(self) -> Tensor:
r"""Get the bounds of each hypercell in the decomposition.
Args:
ref_point: A `(batch_shape) x m`-dim tensor containing the reference point.
Returns:
A `2 x num_cells x m`-dim tensor containing the
lower and upper vertices bounding each hypercell.
"""
ref_point = _expand_ref_point(
ref_point=self.ref_point, batch_shape=self.batch_shape
)
aug_pareto_Y = torch.cat(
[
# -inf is the lower bound of the non-dominated space
torch.full(
torch.Size(
[
*self.batch_shape,
1,
self.num_outcomes,
]
),
float("-inf"),
dtype=self._neg_pareto_Y.dtype,
device=self._neg_pareto_Y.device,
),
self._neg_pareto_Y,
# note: internally, this class minimizes, so use negative here
-(ref_point.unsqueeze(-2)),
],
dim=-2,
)
minimization_cell_bounds = self._get_hypercell_bounds(aug_pareto_Y=aug_pareto_Y)
# swap upper and lower bounds and multiply by -1
return -minimization_cell_bounds.flip(0)
def _get_hypercell_bounds(self, aug_pareto_Y: Tensor) -> Tensor:
r"""Get the bounds of each hypercell in the decomposition.
Args:
aug_pareto_Y: A `n_pareto + 2 x m`-dim tensor containing
the augmented Pareto front.
Returns:
A `2 x (batch_shape) x num_cells x m`-dim tensor containing the
lower and upper vertices bounding each hypercell.
"""
num_cells = self.hypercells.shape[-2]
cells_times_outcomes = num_cells * self.num_outcomes
outcome_idxr = (
torch.arange(self.num_outcomes, dtype=torch.long, device=self._neg_Y.device)
.repeat(num_cells)
.view(
*(1 for _ in self.hypercells.shape[:-2]),
cells_times_outcomes,
)
.expand(*self.hypercells.shape[:-2], cells_times_outcomes)
)
# this tensor is 2 x (num_cells * m) x 2
# the batch dim corresponds to lower/upper bound
cell_bounds_idxr = torch.stack(
[
self.hypercells.view(*self.hypercells.shape[:-2], -1),
outcome_idxr,
],
dim=-1,
).view(2, -1, 2)
if len(self.batch_shape) > 0:
# TODO: support multiple batch dimensions here
batch_idxr = (
torch.arange(
self.batch_shape[0], dtype=torch.long, device=self._neg_Y.device
)
.unsqueeze(1)
.expand(-1, cells_times_outcomes)
.reshape(1, -1, 1)
.expand(2, -1, 1)
)
cell_bounds_idxr = torch.cat([batch_idxr, cell_bounds_idxr], dim=-1)
cell_bounds_values = aug_pareto_Y[
cell_bounds_idxr.chunk(cell_bounds_idxr.shape[-1], dim=-1)
]
view_shape = (2, *self.batch_shape, num_cells, self.num_outcomes)
return cell_bounds_values.view(view_shape)
[docs] def compute_hypervolume(self) -> Tensor:
r"""Compute the hypervolume for the given reference point.
This method computes the hypervolume of the non-dominated space
and computes the difference between the hypervolume between the
ideal point and hypervolume of the non-dominated space.
Returns:
`(batch_shape)`-dim tensor containing the dominated hypervolume.
"""
if self._neg_pareto_Y.shape[-2] == 0:
return torch.zeros(
self._neg_pareto_Y.shape[:-2],
dtype=self._neg_pareto_Y.dtype,
device=self._neg_pareto_Y.device,
)
ref_point = _expand_ref_point(
ref_point=self.ref_point, batch_shape=self.batch_shape
)
# internally we minimize
ref_point = -ref_point.unsqueeze(-2)
ideal_point = self._neg_pareto_Y.min(dim=-2, keepdim=True).values
aug_pareto_Y = torch.cat([ideal_point, self._neg_pareto_Y, ref_point], dim=-2)
cell_bounds_values = self._get_hypercell_bounds(aug_pareto_Y=aug_pareto_Y)
total_volume = (ref_point - ideal_point).squeeze(-2).prod(dim=-1)
non_dom_volume = (
(cell_bounds_values[1] - cell_bounds_values[0]).prod(dim=-1).sum(dim=-1)
)
return total_volume - non_dom_volume
[docs]class FastNondominatedPartitioning(FastPartitioning):
r"""A class for partitioning the non-dominated space into hyper-cells.
Note: this assumes maximization. Internally, it multiplies by -1 and performs
the decomposition under minimization.
This class is far more efficient than NondominatedPartitioning for exact box
partitionings
This class uses the two-step approach similar to that in [Yang2019]_, where:
a) first, Alg 1 from [Lacour17]_ is used to find the local lower bounds
for the maximization problem
b) second, the local lower bounds are used as the Pareto frontier for the
minimization problem, and [Lacour17]_ is applied again to partition
the space dominated by that Pareto frontier.
"""
def __init__(
self,
ref_point: Tensor,
Y: Optional[Tensor] = None,
) -> None:
"""Initialize FastNondominatedPartitioning.
Args:
ref_point: A `m`-dim tensor containing the reference point.
Y: A `(batch_shape) x n x m`-dim tensor.
Example:
>>> bd = FastNondominatedPartitioning(ref_point, Y=Y1)
"""
super().__init__(ref_point=ref_point, Y=Y)
def _get_single_cell(self) -> None:
r"""Set the partitioning to be a single cell in the case of no Pareto points."""
cell_bounds = torch.full(
(2, *self._neg_pareto_Y.shape[:-2], 1, self.num_outcomes),
float("inf"),
dtype=self._neg_pareto_Y.dtype,
device=self._neg_pareto_Y.device,
)
cell_bounds[0] = self.ref_point
self.register_buffer("hypercell_bounds", cell_bounds)
def _get_partitioning(self) -> None:
r"""Compute non-dominated partitioning.
Given local upper bounds for the minimization problem (self._U), this computes
the non-dominated partitioning for the maximization problem. Note that
-self.U contains the local lower bounds for the maximization problem. Following
[Yang2019]_, this treats -self.U as a *new* pareto frontier for a minimization
problem with a reference point of [infinity]^m and computes a dominated
partitioning for this minimization problem.
"""
new_ref_point = torch.full(
torch.Size([1]) + self._neg_ref_point.shape,
float("inf"),
dtype=self._neg_ref_point.dtype,
device=self._neg_ref_point.device,
)
# initialize local upper bounds for the second minimization problem
self.register_buffer("_U2", new_ref_point)
# initialize defining points for the second minimization problem
# use ref point for maximization as the ideal point for minimization.
self._Z2 = self.ref_point.expand(
1, self.num_outcomes, self.num_outcomes
).clone()
for j in range(self._neg_ref_point.shape[-1]):
self._Z2[0, j, j] = self._U2[0, j]
# incrementally update local upper bounds and defining points
# for each new Pareto point
self._U2, self._Z2 = update_local_upper_bounds_incremental(
new_pareto_Y=-self._U,
U=self._U2,
Z=self._Z2,
)
cell_bounds = get_partition_bounds(
Z=self._Z2, U=self._U2, ref_point=new_ref_point.view(-1)
)
self.register_buffer("hypercell_bounds", cell_bounds)
def _partition_space_2d(self) -> None:
r"""Partition the non-dominated space into disjoint hypercells.
This direct method works for `m=2` outcomes.
"""
cell_bounds = compute_non_dominated_hypercell_bounds_2d(
pareto_Y_sorted=self.pareto_Y.flip(-2),
ref_point=self.ref_point,
)
self.register_buffer("hypercell_bounds", cell_bounds)
[docs] def compute_hypervolume(self):
r"""Compute hypervolume that is dominated by the Pareto Froniter.
Returns:
A `(batch_shape)`-dim tensor containing the hypervolume dominated by
each Pareto frontier.
"""
if self._neg_pareto_Y.shape[-2] == 0:
return torch.zeros(
self._neg_pareto_Y.shape[:-2],
dtype=self._neg_pareto_Y.dtype,
device=self._neg_pareto_Y.device,
)
ideal_point = self.pareto_Y.max(dim=-2, keepdim=True).values
total_volume = (
(ideal_point.squeeze(-2) - self.ref_point).clamp_min(0.0).prod(dim=-1)
)
finite_cell_bounds = torch.min(self.hypercell_bounds, ideal_point)
non_dom_volume = (
(finite_cell_bounds[1] - finite_cell_bounds[0])
.clamp_min(0.0)
.prod(dim=-1)
.sum(dim=-1)
)
return total_volume - non_dom_volume
