Error Bounds for Sequential Monte Carlo Samplers for Multimodal Distributions

TL;DR

This paper derives bounds on the asymptotic variance of SMC samplers for multimodal distributions, demonstrating improved efficiency and applicability, especially in challenging models like the Potts model.

Contribution

It introduces new variance bounds for SMC methods that incorporate MCMC kernels capable of moving between modes, and proposes an interpolation approach for efficient sampling.

Findings

Bounds on asymptotic variance are improved over previous results.

SMC with interpolation can sample multimodal distributions at polynomial cost.

Application to the Potts model shows practical efficiency gains.

Abstract

In this paper, we provide bounds on the asymptotic variance for a class of sequential Monte Carlo (SMC) samplers designed for approximating multimodal distributions. Such methods combine standard SMC methods and Markov chain Monte Carlo (MCMC) kernels. Our bounds improve upon previous results, and unlike some earlier work, they also apply in the case when the MCMC kernels can move between the modes. We apply our results to the Potts model from statistical physics. In this case, the problem of sharp peaks is encountered. Earlier methods, such as parallel tempering, are only able to sample from it at an exponential (in an important parameter of the model) cost. We propose a sequence of interpolating distributions called interpolation to independence, and show that the SMC sampler based on it is able to sample from this target distribution at a polynomial cost. We believe that our method is generally applicable to many other distributions as well.