Convergence Bounds for Sequential Monte Carlo on Multimodal Distributions using Soft Decomposition

TL;DR

This paper establishes variance bounds for Sequential Monte Carlo algorithms applied to multimodal distributions, demonstrating that local mixing properties suffice for theoretical guarantees, unlike traditional global bounds.

Contribution

It introduces new variance bounds for SMC that depend on local rather than global mixing times, improving theoretical understanding for multimodal distributions.

Findings

Variance bounds depend on local MCMC mixing times.

SMC can effectively handle multimodal distributions.

Theoretical guarantees extend beyond uni-modal settings.

Abstract

We prove bounds on the variance of a function $f$ under the empirical measure of the samples obtained by the Sequential Monte Carlo (SMC) algorithm, with time complexity depending on local rather than global Markov chain mixing dynamics. SMC is a Markov Chain Monte Carlo (MCMC) method, which starts by drawing $N$ particles from a known distribution, and then, through a sequence of distributions, re-weights and re-samples the particles, at each instance applying a Markov chain for smoothing. In principle, SMC tries to alleviate problems from multi-modality. However, most theoretical guarantees for SMC are obtained by assuming global mixing time bounds, which are only efficient in the uni-modal setting. We show that bounds can be obtained in the truly multi-modal setting, with mixing times that depend only on local MCMC dynamics.