#!/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"""
Utilities for MC and qMC sampling.
References
.. [Trikalinos2014polytope]
T. A. Trikalinos and G. van Valkenhoef. Efficient sampling from uniform
density n-polytopes. Technical report, Brown University, 2014.
"""
from __future__ import annotations
import warnings
from abc import ABC, abstractmethod
from contextlib import contextmanager
from typing import Generator, Iterable, List, Optional, Tuple
import numpy as np
import scipy
import torch
from botorch.exceptions.errors import BotorchError
from botorch.exceptions.warnings import SamplingWarning
from botorch.posteriors.posterior import Posterior
from botorch.sampling.qmc import NormalQMCEngine
from scipy.spatial import Delaunay, HalfspaceIntersection
from torch import LongTensor, Tensor
from torch.quasirandom import SobolEngine
[docs]@contextmanager
def manual_seed(seed: Optional[int] = None) -> Generator[None, None, None]:
r"""Contextmanager for manual setting the torch.random seed.
Args:
seed: The seed to set the random number generator to.
Returns:
Generator
Example:
>>> with manual_seed(1234):
>>> X = torch.rand(3)
"""
old_state = torch.random.get_rng_state()
try:
if seed is not None:
torch.random.manual_seed(seed)
yield
finally:
if seed is not None:
torch.random.set_rng_state(old_state)
[docs]def construct_base_samples(
batch_shape: torch.Size,
output_shape: torch.Size,
sample_shape: torch.Size,
qmc: bool = True,
seed: Optional[int] = None,
device: Optional[torch.device] = None,
dtype: Optional[torch.dtype] = None,
) -> Tensor:
r"""Construct base samples from a multi-variate standard normal N(0, I_qo).
Args:
batch_shape: The batch shape of the base samples to generate. Typically,
this is used with each dimension of size 1, so as to eliminate
sampling variance across batches.
output_shape: The output shape (`q x m`) of the base samples to generate.
sample_shape: The sample shape of the samples to draw.
qmc: If True, use quasi-MC sampling (instead of iid draws).
seed: If provided, use as a seed for the RNG.
Returns:
A `sample_shape x batch_shape x mutput_shape` dimensional tensor of base
samples, drawn from a N(0, I_qm) distribution (using QMC if `qmc=True`).
Here `output_shape = q x m`.
Example:
>>> batch_shape = torch.Size([2])
>>> output_shape = torch.Size([3])
>>> sample_shape = torch.Size([10])
>>> samples = construct_base_samples(batch_shape, output_shape, sample_shape)
"""
base_sample_shape = batch_shape + output_shape
output_dim = output_shape.numel()
if qmc and output_dim <= SobolEngine.MAXDIM:
n = (sample_shape + batch_shape).numel()
base_samples = draw_sobol_normal_samples(
d=output_dim, n=n, device=device, dtype=dtype, seed=seed
)
base_samples = base_samples.view(sample_shape + base_sample_shape)
else:
if qmc and output_dim > SobolEngine.MAXDIM:
warnings.warn(
f"Number of output elements (q*d={output_dim}) greater than "
f"maximum supported by qmc ({SobolEngine.MAXDIM}). "
"Using iid sampling instead.",
SamplingWarning,
)
with manual_seed(seed=seed):
base_samples = torch.randn(
sample_shape + base_sample_shape, device=device, dtype=dtype
)
return base_samples
[docs]def construct_base_samples_from_posterior(
posterior: Posterior,
sample_shape: torch.Size,
qmc: bool = True,
collapse_batch_dims: bool = True,
seed: Optional[int] = None,
) -> Tensor:
r"""Construct a tensor of normally distributed base samples.
Args:
posterior: A Posterior object.
sample_shape: The sample shape of the samples to draw.
qmc: If True, use quasi-MC sampling (instead of iid draws).
seed: If provided, use as a seed for the RNG.
Returns:
A `num_samples x 1 x q x m` dimensional Tensor of base samples, drawn
from a N(0, I_qm) distribution (using QMC if `qmc=True`). Here `q` and
`m` are the same as in the posterior's `event_shape` `b x q x m`.
Importantly, this only obtain a single t-batch of samples, so as to not
introduce any sampling variance across t-batches.
Example:
>>> sample_shape = torch.Size([10])
>>> samples = construct_base_samples_from_posterior(posterior, sample_shape)
"""
output_shape = posterior.event_shape[-2:] # shape of joint output: q x m
if collapse_batch_dims:
batch_shape = torch.Size([1] * len(posterior.event_shape[:-2]))
else:
batch_shape = posterior.event_shape[:-2]
base_samples = construct_base_samples(
batch_shape=batch_shape,
output_shape=output_shape,
sample_shape=sample_shape,
qmc=qmc,
seed=seed,
device=posterior.device,
dtype=posterior.dtype,
)
return base_samples
[docs]def draw_sobol_samples(
bounds: Tensor,
n: int,
q: int,
batch_shape: Optional[Iterable[int], torch.Size] = None,
seed: Optional[int] = None,
) -> Tensor:
r"""Draw qMC samples from the box defined by bounds.
Args:
bounds: A `2 x d` dimensional tensor specifying box constraints on a
`d`-dimensional space, where bounds[0, :] and bounds[1, :] correspond
to lower and upper bounds, respectively.
n: The number of (q-batch) samples. As a best practice, use powers of 2.
q: The size of each q-batch.
batch_shape: The batch shape of the samples. If given, returns samples
of shape `n x batch_shape x q x d`, where each batch is an
`n x q x d`-dim tensor of qMC samples.
seed: The seed used for initializing Owen scrambling. If None (default),
use a random seed.
Returns:
A `n x batch_shape x q x d`-dim tensor of qMC samples from the box
defined by bounds.
Example:
>>> bounds = torch.stack([torch.zeros(3), torch.ones(3)])
>>> samples = draw_sobol_samples(bounds, 16, 2)
"""
batch_shape = batch_shape or torch.Size()
batch_size = int(torch.prod(torch.tensor(batch_shape)))
d = bounds.shape[-1]
lower = bounds[0]
rng = bounds[1] - bounds[0]
sobol_engine = SobolEngine(q * d, scramble=True, seed=seed)
samples_raw = sobol_engine.draw(batch_size * n, dtype=lower.dtype)
samples_raw = samples_raw.view(*batch_shape, n, q, d).to(device=lower.device)
if batch_shape != torch.Size():
samples_raw = samples_raw.permute(-3, *range(len(batch_shape)), -2, -1)
return lower + rng * samples_raw
[docs]def draw_sobol_normal_samples(
d: int,
n: int,
device: Optional[torch.device] = None,
dtype: Optional[torch.dtype] = None,
seed: Optional[int] = None,
) -> Tensor:
r"""Draw qMC samples from a multi-variate standard normal N(0, I_d)
A primary use-case for this functionality is to compute an QMC average
of f(X) over X where each element of X is drawn N(0, 1).
Args:
d: The dimension of the normal distribution.
n: The number of samples to return. As a best practice, use powers of 2.
device: The torch device.
dtype: The torch dtype.
seed: The seed used for initializing Owen scrambling. If None (default),
use a random seed.
Returns:
A tensor of qMC standard normal samples with dimension `n x d` with device
and dtype specified by the input.
Example:
>>> samples = draw_sobol_normal_samples(2, 16)
"""
normal_qmc_engine = NormalQMCEngine(d=d, seed=seed, inv_transform=True)
samples = normal_qmc_engine.draw(n, dtype=torch.float if dtype is None else dtype)
return samples.to(device=device)
[docs]def sample_hypersphere(
d: int,
n: int = 1,
qmc: bool = False,
seed: Optional[int] = None,
device: Optional[torch.device] = None,
dtype: Optional[torch.dtype] = None,
) -> Tensor:
r"""Sample uniformly from a unit d-sphere.
Args:
d: The dimension of the hypersphere.
n: The number of samples to return.
qmc: If True, use QMC Sobol sampling (instead of i.i.d. uniform).
seed: If provided, use as a seed for the RNG.
device: The torch device.
dtype: The torch dtype.
Returns:
An `n x d` tensor of uniform samples from from the d-hypersphere.
Example:
>>> sample_hypersphere(d=5, n=10)
"""
dtype = torch.float if dtype is None else dtype
if d == 1:
rnd = torch.randint(0, 2, (n, 1), device=device, dtype=dtype)
return 2 * rnd - 1
if qmc:
rnd = draw_sobol_normal_samples(d=d, n=n, device=device, dtype=dtype, seed=seed)
else:
with manual_seed(seed=seed):
rnd = torch.randn(n, d, dtype=dtype)
samples = rnd / torch.norm(rnd, dim=-1, keepdim=True)
if device is not None:
samples = samples.to(device)
return samples
[docs]def sample_simplex(
d: int,
n: int = 1,
qmc: bool = False,
seed: Optional[int] = None,
device: Optional[torch.device] = None,
dtype: Optional[torch.dtype] = None,
) -> Tensor:
r"""Sample uniformly from a d-simplex.
Args:
d: The dimension of the simplex.
n: The number of samples to return.
qmc: If True, use QMC Sobol sampling (instead of i.i.d. uniform).
seed: If provided, use as a seed for the RNG.
device: The torch device.
dtype: The torch dtype.
Returns:
An `n x d` tensor of uniform samples from from the d-simplex.
Example:
>>> sample_simplex(d=3, n=10)
"""
dtype = torch.float if dtype is None else dtype
if d == 1:
return torch.ones(n, 1, device=device, dtype=dtype)
if qmc:
sobol_engine = SobolEngine(d - 1, scramble=True, seed=seed)
rnd = sobol_engine.draw(n, dtype=dtype)
else:
with manual_seed(seed=seed):
rnd = torch.rand(n, d - 1, dtype=dtype)
srnd, _ = torch.sort(rnd, dim=-1)
zeros = torch.zeros(n, 1, dtype=dtype)
ones = torch.ones(n, 1, dtype=dtype)
srnd = torch.cat([zeros, srnd, ones], dim=-1)
if device is not None:
srnd = srnd.to(device)
return srnd[..., 1:] - srnd[..., :-1]
[docs]def sample_polytope(
A: Tensor,
b: Tensor,
x0: Tensor,
n: int = 10000,
n0: int = 100,
seed: Optional[int] = None,
) -> Tensor:
r"""
Hit and run sampler from uniform sampling points from a polytope,
described via inequality constraints A*x<=b.
Args:
A: A Tensor describing inequality constraints
so that all samples satisfy Ax<=b.
b: A Tensor describing the inequality constraints
so that all samples satisfy Ax<=b.
x0: A `d`-dim Tensor representing a starting point of the chain
satisfying the constraints.
n: The number of resulting samples kept in the output.
n0: The number of burn-in samples. The chain will produce
n+n0 samples but the first n0 samples are not saved.
seed: The seed for the sampler. If omitted, use a random seed.
Returns:
(n, d) dim Tensor containing the resulting samples.
"""
n_tot = n + n0
seed = seed if seed is not None else torch.randint(0, 1000000, (1,)).item()
with manual_seed(seed=seed):
rands = torch.rand(n_tot, dtype=A.dtype, device=A.device)
# pre-sample samples from hypersphere
d = x0.size(0)
# uniform samples from unit ball in d dims
Rs = sample_hypersphere(d=d, n=n_tot, dtype=A.dtype, device=A.device).unsqueeze(-1)
# compute matprods in batch
ARs = (A @ Rs).squeeze(-1)
out = torch.empty(n, A.size(-1), dtype=A.dtype, device=A.device)
x = x0.clone()
for i, (ar, r, rnd) in enumerate(zip(ARs, Rs, rands)):
# given x, the next point in the chain is x+alpha*r
# it also satisfies A(x+alpha*r)<=b which implies A*alpha*r<=b-Ax
# so alpha<=(b-Ax)/ar for ar>0, and alpha>=(b-Ax)/ar for ar<0.
# b - A @ x is always >= 0, clamping for numerical tolerances
w = (b - A @ x).squeeze().clamp(min=0.0) / ar
pos = w >= 0
alpha_max = w[pos].min()
# important to include equality here in cases x is at the boundary
# of the polytope
neg = w <= 0
alpha_min = w[neg].max()
# alpha~Unif[alpha_min, alpha_max]
alpha = alpha_min + rnd * (alpha_max - alpha_min)
x = x + alpha * r
if i >= n0: # save samples after burn-in period
out[i - n0] = x.squeeze()
return out
[docs]def batched_multinomial(
weights: Tensor,
num_samples: int,
replacement: bool = False,
generator: Optional[torch.Generator] = None,
out: Optional[Tensor] = None,
) -> LongTensor:
r"""Sample from multinomial with an arbitrary number of batch dimensions.
Args:
weights: A `batch_shape x num_categories` tensor of weights. For each batch
index `i, j, ...`, this functions samples from a multinomial with `input`
`weights[i, j, ..., :]`. Note that the weights need not sum to one, but must
be non-negative, finite and have a non-zero sum.
num_samples: The number of samples to draw for each batch index. Must be smaller
than `num_categories` if `replacement=False`.
replacement: If True, samples are drawn with replacement.
generator: A a pseudorandom number generator for sampling.
out: The output tensor (optional). If provided, must be of size
`batch_shape x num_samples`.
Returns:
A `batch_shape x num_samples` tensor of samples.
This is a thin wrapper around `torch.multinomial` that allows weight (`input`)
tensors with an arbitrary number of batch dimensions (`torch.multinomial` only
allows a single batch dimension). The calling signature is the same as for
`torch.multinomial`.
Example:
>>> weights = torch.rand(2, 3, 10)
>>> samples = batched_multinomial(weights, 4) # shape is 2 x 3 x 4
"""
batch_shape, n_categories = weights.shape[:-1], weights.size(-1)
flat_samples = torch.multinomial(
input=weights.view(-1, n_categories),
num_samples=num_samples,
replacement=replacement,
generator=generator,
out=None if out is None else out.view(-1, num_samples),
)
return flat_samples.view(*batch_shape, num_samples)
def _convert_bounds_to_inequality_constraints(bounds: Tensor) -> Tuple[Tensor, Tensor]:
r"""Convert bounds into inequality constraints of the form Ax <= b.
Args:
bounds: A `2 x d`-dim tensor of bounds
Returns:
A two-element tuple containing
- A: A `2d x d`-dim tensor of coefficients
- b: A `2d x 1`-dim tensor containing the right hand side
"""
d = bounds.shape[-1]
eye = torch.eye(d, dtype=bounds.dtype, device=bounds.device)
lower, upper = bounds
lower_finite, upper_finite = bounds.isfinite()
A = torch.cat([-eye[lower_finite], eye[upper_finite]], dim=0)
b = torch.cat([-lower[lower_finite], upper[upper_finite]], dim=0).unsqueeze(-1)
return A, b
[docs]def find_interior_point(
A: np.ndarray,
b: np.ndarray,
A_eq: Optional[np.ndarray] = None,
b_eq: Optional[np.ndarray] = None,
) -> np.ndarray:
r"""Find an interior point of a polytope via linear programming.
Args:
A: A `n_ineq x d`-dim numpy array containing the coefficients of the
constraint inequalities.
b: A `n_ineq x 1`-dim numpy array containing the right hand sides of
the constraint inequalities.
A_eq: A `n_eq x d`-dim numpy array containing the coefficients of the
constraint equalities.
b_eq: A `n_eq x 1`-dim numpy array containing the right hand sides of
the constraint equalities.
Returns:
A `d`-dim numpy array containing an interior point of the polytope.
This function will raise a ValueError if there is no such point.
This method solves the following Linear Program:
min -s subject to A @ x <= b - 2 * s, s >= 0, A_eq @ x = b_eq
In case the polytope is unbounded, then it will also constrain the slack
variable `s` to `s<=1`.
"""
# augment inequality constraints: A @ (x, s) <= b
d = A.shape[-1]
ncon = A.shape[-2] + 1
c = np.zeros(d + 1)
c[-1] = -1
b_ub = np.zeros(ncon)
b_ub[:-1] = b.reshape(-1)
A_ub = np.zeros((ncon, d + 1))
A_ub[:-1, :-1] = A
A_ub[:-1, -1] = 2.0
A_ub[-1, -1] = -1.0
result = scipy.optimize.linprog(
c=c, A_ub=A_ub, b_ub=b_ub, A_eq=A_eq, b_eq=b_eq, bounds=(None, None)
)
if result.status == 3:
# problem is unbounded - to find a bounded solution we constrain the
# slack variable `s` to `s <= 1.0`.
A_s = np.concatenate([np.zeros((1, d)), np.ones((1, 1))], axis=-1)
A_ub = np.concatenate([A_ub, A_s], axis=0)
b_ub = np.concatenate([b_ub, np.ones(1)], axis=-1)
result = scipy.optimize.linprog(
c=c, A_ub=A_ub, b_ub=b_ub, A_eq=A_eq, b_eq=b_eq, bounds=(None, None)
)
if result.status == 2:
raise ValueError(
"No feasible point found. Constraint polytope appears empty. "
+ "Check your constraints."
)
elif result.status > 0:
raise ValueError(
"Problem checking constraint specification. "
+ "linprog status: {}".format(result.message)
)
# the x in the result is really (x, s)
return result.x[:-1]
[docs]class PolytopeSampler(ABC):
r"""Base class for samplers that sample points from a polytope."""
def __init__(
self,
inequality_constraints: Optional[Tuple[Tensor, Tensor]] = None,
equality_constraints: Optional[Tuple[Tensor, Tensor]] = None,
bounds: Optional[Tensor] = None,
interior_point: Optional[Tensor] = None,
) -> None:
r"""Initialize PolytopeSampler.
Args:
inequality_constraints: Tensors `(A, b)` describing inequality
constraints `A @ x <= b`, where `A` is a `n_ineq_con x d`-dim
Tensor and `b` is a `n_ineq_con x 1`-dim Tensor, with `n_ineq_con`
the number of inequalities and `d` the dimension of the sample space.
equality_constraints: Tensors `(C, d)` describing the equality constraints
`C @ x = d`, where `C` is a `n_eq_con x d`-dim Tensor and `d` is a
`n_eq_con x 1`-dim Tensor with `n_eq_con` the number of equalities.
bounds: A `2 x d`-dim tensor of box bounds, where `inf` (`-inf`) means
that the respective dimension is unbounded above (below).
interior_point: A `d x 1`-dim Tensor presenting a point in the
(relative) interior of the polytope. If omitted, determined
automatically by solving a Linear Program.
"""
if inequality_constraints is None:
if bounds is None:
raise BotorchError(
"PolytopeSampler requires either inequality constraints or bounds."
)
A = torch.empty(
0, bounds.shape[-1], dtype=bounds.dtype, device=bounds.device
)
b = torch.empty(0, 1, dtype=bounds.dtype, device=bounds.device)
else:
A, b = inequality_constraints
if bounds is not None:
# add inequality constraints for bounds
# TODO: make sure there are not deduplicate constraints
A2, b2 = _convert_bounds_to_inequality_constraints(bounds=bounds)
A = torch.cat([A, A2], dim=0)
b = torch.cat([b, b2], dim=0)
self.A = A
self.b = b
self.equality_constraints = equality_constraints
if equality_constraints is not None:
self.C, self.d = equality_constraints
U, S, Vh = torch.linalg.svd(self.C)
r = torch.nonzero(S).size(0) # rank of matrix C
self.nullC = Vh[r:, :].transpose(-1, -2) # orthonormal null space of C,
# satisfying # C @ nullC = 0 and nullC.T @ nullC = I
# using the change of variables x=x0+nullC*y,
# sample y satisfies A*nullC*y<=b-A*x0.
# the linear constraint is automatically satisfied as x0 satisfies it.
else:
self.C = None
self.d = None
self.nullC = torch.eye(
self.A.size(-1), dtype=self.A.dtype, device=self.A.device
)
self.new_A = self.A @ self.nullC # doesn't depend on the initial point
# initial point for the original, not transformed, problem
if interior_point is not None:
if self.feasible(interior_point):
self.x0 = interior_point
else:
raise ValueError("The given input point is not feasible.")
else:
self.x0 = self.find_interior_point()
[docs] def feasible(self, x: Tensor) -> bool:
r"""Check whether a point is contained in the polytope.
Args:
x: A `d x 1`-dim Tensor.
Returns:
True if `x` is contained inside the polytope (incl. its boundary),
False otherwise.
"""
ineq = (self.A @ x - self.b <= 0).all()
if self.equality_constraints is not None:
eq = (self.C @ x - self.d == 0).all()
return ineq & eq
return ineq
[docs] def find_interior_point(self) -> Tensor:
r"""Find an interior point of the polytope.
Returns:
A `d x 1`-dim Tensor representing a point contained in the polytope.
This function will raise a ValueError if there is no such point.
"""
if self.equality_constraints:
# equality constraints: A_eq * (x, s) = b_eq
A_eq = np.zeros((self.C.size(0), self.C.size(-1) + 1))
A_eq[:, :-1] = self.C.cpu().numpy()
b_eq = self.d.cpu().numpy()
else:
A_eq = None
b_eq = None
x0 = find_interior_point(
A=self.A.cpu().numpy(), b=self.b.cpu().numpy(), A_eq=A_eq, b_eq=b_eq
)
return torch.from_numpy(x0).to(self.A).unsqueeze(-1)
# -------- Abstract methods to be implemented by subclasses -------- #
[docs] @abstractmethod
def draw(self, n: int = 1, seed: Optional[int] = None) -> Tensor:
r"""Draw samples from the polytope.
Args:
n: The number of samples.
seed: The random seed.
Returns:
A `n x d` Tensor of samples from the polytope.
"""
pass # pragma: no cover
[docs]class HitAndRunPolytopeSampler(PolytopeSampler):
r"""A sampler for sampling from a polyope using a hit-and-run algorithm."""
def __init__(
self,
inequality_constraints: Optional[Tuple[Tensor, Tensor]] = None,
equality_constraints: Optional[Tuple[Tensor, Tensor]] = None,
bounds: Optional[Tensor] = None,
interior_point: Optional[Tensor] = None,
n_burnin: int = 0,
) -> None:
r"""A sampler for sampling from a polyope using a hit-and-run algorithm.
Args:
inequality_constraints: Tensors `(A, b)` describing inequality
constraints `A @ x <= b`, where `A` is a `n_ineq_con x d`-dim
Tensor and `b` is a `n_ineq_con x 1`-dim Tensor, with `n_ineq_con`
the number of inequalities and `d` the dimension of the sample space.
equality_constraints: Tensors `(C, d)` describing the equality constraints
`C @ x = d`, where `C` is a `n_eq_con x d`-dim Tensor and `d` is a
`n_eq_con x 1`-dim Tensor with `n_eq_con` the number of equalities.
bounds: A `2 x d`-dim tensor of box bounds, where `inf` (`-inf`) means
that the respective dimension is unbounded from above (below).
interior_point: A `d x 1`-dim Tensor representing a point in the
(relative) interior of the polytope. If omitted, determined
automatically by solving a Linear Program.
n_burnin: The number of burn in samples.
"""
super().__init__(
inequality_constraints=inequality_constraints,
equality_constraints=equality_constraints,
bounds=bounds,
interior_point=interior_point,
)
self.n_burnin = n_burnin
[docs] def draw(self, n: int = 1, seed: Optional[int] = None) -> Tensor:
r"""Draw samples from the polytope.
Args:
n: The number of samples.
seed: The random seed.
Returns:
A `n x d` Tensor of samples from the polytope.
"""
transformed_samples = sample_polytope(
# run this on the cpu
A=self.new_A.cpu(),
b=(self.b - self.A @ self.x0).cpu(),
x0=torch.zeros((self.nullC.size(1), 1), dtype=self.A.dtype),
n=n,
n0=self.n_burnin,
seed=seed,
).to(self.b)
init_shift = self.x0.transpose(-1, -2)
samples = init_shift + transformed_samples @ self.nullC.transpose(-1, -2)
# keep the last element of the resulting chain as
# the beginning of the next chain
self.x0 = samples[-1].reshape(-1, 1)
# reset counter so there is no burn-in for subsequent samples
self.n_burnin = 0
return samples
[docs]class DelaunayPolytopeSampler(PolytopeSampler):
r"""A polytope sampler using Delaunay triangulation.
This sampler first enumerates the vertices of the constraint polytope and
then uses a Delaunay triangulation to tesselate its convex hull.
The sampling happens in two stages:
1. First, we sample from the set of hypertriangles generated by the
Delaunay triangulation (i.e. which hyper-triangle to draw the sample
from) with probabilities proportional to the triangle volumes.
2. Then, we sample uniformly from the chosen hypertriangle by sampling
uniformly from the unit simplex of the appropriate dimension, and
then computing the convex combination of the vertices of the
hypertriangle according to that draw from the simplex.
The best reference (not exactly the same, but functionally equivalent) is
[Trikalinos2014polytope]_. A simple R implementation is available at
https://github.com/gertvv/tesselample.
"""
def __init__(
self,
inequality_constraints: Optional[Tuple[Tensor, Tensor]] = None,
equality_constraints: Optional[Tuple[Tensor, Tensor]] = None,
bounds: Optional[Tensor] = None,
interior_point: Optional[Tensor] = None,
) -> None:
r"""Initialize DelaunayPolytopeSampler.
Args:
inequality_constraints: Tensors `(A, b)` describing inequality
constraints `A @ x <= b`, where `A` is a `n_ineq_con x d`-dim
Tensor and `b` is a `n_ineq_con x 1`-dim Tensor, with `n_ineq_con`
the number of inequalities and `d` the dimension of the sample space.
equality_constraints: Tensors `(C, d)` describing the equality constraints
`C @ x = d`, where `C` is a `n_eq_con x d`-dim Tensor and `d` is a
`n_eq_con x 1`-dim Tensor with `n_eq_con` the number of equalities.
bounds: A `2 x d`-dim tensor of box bounds, where `inf` (`-inf`) means
that the respective dimension is unbounded from above (below).
interior_point: A `d x 1`-dim Tensor representing a point in the
(relative) interior of the polytope. If omitted, determined
automatically by solving a Linear Program.
Warning: The vertex enumeration performed in this algorithm can become
extremely costly if there are a large number of inequalities. Similarly,
the triangulation can get very expensive in high dimensions. Only use
this algorithm for moderate dimensions / moderately complex constraint sets.
An alternative is the `HitAndRunPolytopeSampler`.
"""
super().__init__(
inequality_constraints=inequality_constraints,
equality_constraints=equality_constraints,
bounds=bounds,
interior_point=interior_point,
)
# shift coordinate system to be anchored at x0
new_b = self.b - self.A @ self.x0
if self.new_A.shape[-1] < 2:
# if the polytope is in dim 1 (i.e. a line segment) Qhull won't work
tshlds = new_b / self.new_A
neg = self.new_A < 0
self.y_min = tshlds[neg].max()
self.y_max = tshlds[~neg].min()
self.dim = 1
else:
# Qhull expects inputs of the form A @ x + b <= 0, so we need to negate here
halfspaces = torch.cat([self.new_A, -new_b], dim=-1).cpu().numpy()
vertices = HalfspaceIntersection(
halfspaces=halfspaces, interior_point=np.zeros(self.new_A.shape[-1])
).intersections
self.dim = vertices.shape[-1]
try:
delaunay = Delaunay(vertices)
except ValueError as e:
if "Points cannot contain NaN" in str(e):
raise ValueError("Polytope is unbounded.")
raise e # pragma: no cover
polytopes = torch.from_numpy(
np.array([delaunay.points[s] for s in delaunay.simplices]),
).to(self.A)
volumes = torch.stack([torch.det(p[1:] - p[0]).abs() for p in polytopes])
self._polytopes = polytopes
self._p = volumes / volumes.sum()
[docs] def draw(self, n: int = 1, seed: Optional[int] = None) -> Tensor:
r"""Draw samples from the polytope.
Args:
n: The number of samples.
seed: The random seed.
Returns:
A `n x d` Tensor of samples from the polytope.
"""
if self.dim == 1:
with manual_seed(seed):
e = torch.rand(n, 1, device=self.new_A.device, dtype=self.new_A.dtype)
transformed_samples = self.y_min + (self.y_max - self.y_min) * e
else:
if seed is None:
generator = None
else:
generator = torch.Generator(device=self.A.device)
generator.manual_seed(seed)
index_rvs = torch.multinomial(
self._p,
num_samples=n,
replacement=True,
generator=generator,
)
simplex_rvs = sample_simplex(
d=self.dim + 1, n=n, seed=seed, device=self.A.device, dtype=self.A.dtype
)
transformed_samples = torch.stack(
[rv @ self._polytopes[idx] for rv, idx in zip(simplex_rvs, index_rvs)]
)
init_shift = self.x0.transpose(-1, -2)
samples = init_shift + transformed_samples @ self.nullC.transpose(-1, -2)
return samples
[docs]def get_polytope_samples(
n: int,
bounds: Tensor,
inequality_constraints: Optional[List[Tuple[Tensor, Tensor, float]]] = None,
equality_constraints: Optional[List[Tuple[Tensor, Tensor, float]]] = None,
seed: Optional[int] = None,
thinning: int = 32,
n_burnin: int = 10_000,
) -> Tensor:
r"""Sample from polytope defined by box bounds and (in)equality constraints.
This uses a hit-and-run Markov chain sampler.
TODO: make this method return the sampler object, to avoid doing burn-in
every time we draw samples.
Args:
n: The number of samples.
bounds: A `2 x d`-dim tensor containing the box bounds.
inequality constraints: A list of tuples (indices, coefficients, rhs),
with each tuple encoding an inequality constraint of the form
`\sum_i (X[indices[i]] * coefficients[i]) >= rhs`.
equality constraints: A list of tuples (indices, coefficients, rhs),
with each tuple encoding an inequality constraint of the form
`\sum_i (X[indices[i]] * coefficients[i]) = rhs`.
seed: The random seed.
thinning: The amount of thinning.
n_burnin: The number of burn-in samples for the Markov chain sampler.
Returns:
A `n x d`-dim tensor of samples.
"""
# create tensors representing linear inequality constraints
# of the form Ax >= b.
if inequality_constraints:
A, b = sparse_to_dense_constraints(
d=bounds.shape[-1],
constraints=inequality_constraints,
)
# Note the inequality constraints are of the form Ax >= b,
# but PolytopeSampler expects inequality constraints of the
# form Ax <= b, so we multiply by -1 below.
dense_inequality_constraints = -A, -b
else:
dense_inequality_constraints = None
if equality_constraints:
dense_equality_constraints = sparse_to_dense_constraints(
d=bounds.shape[-1],
constraints=equality_constraints,
)
else:
dense_equality_constraints = None
polytope_sampler = HitAndRunPolytopeSampler(
inequality_constraints=dense_inequality_constraints,
bounds=bounds,
equality_constraints=dense_equality_constraints,
n_burnin=n_burnin,
)
return polytope_sampler.draw(n=n * thinning, seed=seed)[::thinning]
[docs]def sparse_to_dense_constraints(
d: int,
constraints: List[Tuple[Tensor, Tensor, float]],
) -> Tuple[Tensor, Tensor]:
r"""Convert parameter constraints from a sparse format into a dense format.
This method converts sparse triples of the form (indices, coefficients, rhs)
to constraints of the form Ax >= b or Ax = b.
Args:
d: The input dimension.
inequality constraints: A list of tuples (indices, coefficients, rhs),
with each tuple encoding an (in)equality constraint of the form
`\sum_i (X[indices[i]] * coefficients[i]) >= rhs` or
`\sum_i (X[indices[i]] * coefficients[i]) = rhs`.
Returns:
A two-element tuple containing:
- A: A `n_constraints x d`-dim tensor of coefficients.
- b: A `n_constraints x 1`-dim tensor of right hand sides.
"""
_t = constraints[0][1]
A = torch.zeros(len(constraints), d, dtype=_t.dtype, device=_t.device)
b = torch.zeros(len(constraints), 1, dtype=_t.dtype, device=_t.device)
for i, (indices, coefficients, rhs) in enumerate(constraints):
A[i, indices.long()] = coefficients
b[i] = rhs
return A, b
