Sampling from Binary Quadratic Distributions via Stochastic Localization

TL;DR

This paper introduces a novel application of stochastic localization to binary quadratic distributions, providing theoretical guarantees for efficient sampling and demonstrating improved practical sampling performance in complex discrete optimization problems.

Contribution

First to apply stochastic localization to BQDs, proving polynomial-time mixing and broad applicability to various discrete MCMC samplers with theoretical and experimental validation.

Findings

Proves posterior distributions satisfy Poincaré inequalities after SL

Demonstrates polynomial-time mixing for BQDs using SL

Shows improved sampling efficiency in maximum independent set, cut, and clique problems

Abstract

Sampling from binary quadratic distributions (BQDs) is a fundamental but challenging problem in discrete optimization and probabilistic inference. Previous work established theoretical guarantees for stochastic localization (SL) in continuous domains, where MCMC methods efficiently estimate the required posterior expectations during SL iterations. However, achieving similar convergence guarantees for discrete MCMC samplers in posterior estimation presents unique theoretical challenges. In this work, we present the first application of SL to general BQDs, proving that after a certain number of iterations, the external field of posterior distributions constructed by SL tends to infinity almost everywhere, hence satisfy Poincar\'e inequalities with probability near to 1, leading to polynomial-time mixing. This theoretical breakthrough enables efficient sampling from general BQDs, even those that may not originally possess fast mixing properties. Furthermore, our analysis, covering enormous discrete MCMC samplers based on Glauber dynamics and Metropolis-Hastings algorithms, demonstrates the broad applicability of our theoretical framework. Experiments on instances with quadratic unconstrained binary objectives, including maximum independent set, maximum cut, and maximum clique problems, demonstrate consistent improvements in sampling efficiency across different discrete MCMC samplers.