Piecewise Deterministic Sampling for Constrained Distributions

TL;DR

This paper introduces a new class of Piecewise Deterministic Markov Processes (PDMPs) for efficient, unbiased sampling from constrained convex distributions, outperforming existing SDE-based methods.

Contribution

It adapts mirror maps from convex optimization to constrained sampling, enabling unbiased, exact subsampling and improved performance over prior SDE-based algorithms.

Findings

Algorithms outperform state-of-the-art SDE-based methods in constrained sampling tasks.

Proposed PDMPs provide unbiased samples that respect constraints and allow exact subsampling.

Demonstrated advantages across various constrained sampling problems.

Abstract

In this paper, we propose a novel class of Piecewise Deterministic Markov Processes (PDMPs) that are designed to sample from probability distributions $\pi$ supported on a convex set $\mathcal{M}$. This class of PDMPs adapts the concept of a mirror map from convex optimisation to address sampling problems. The corresponding algorithms provide unbiased samples that respect the constraints and, moreover, allow for exact subsampling. We demonstrate the advantages of these algorithms against a range of constrained sampling problems where the proposed algorithms outperform state of the art stochastic differential equation-based methods.