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

Holden Lee, Matheau Santana-Gijzen

TL;DR

The paper addresses the challenge of proving non-asymptotic variance bounds for Sequential Monte Carlo (SMC) when targeting multimodal distributions where global mixing is inefficient.It develops a framework that bounds variance using local mixing dynamics by decomposing the target measure into mixture components and imposing a generator-decomposition and log-Sobolev constraints, plus a density-ratio bound across levels.Key contributions include a detailed non-asymptotic variance bound, high-probability and total-variation guarantees, and verifiable applicability to Langevin diffusion, Glauber dynamics, Metropolis-Hastings, and TRE-based sampling approaches.The results provide theoretical justification for SMC’s empirical effectiveness in multimodal settings and offer guidance on choosing sampling times and particle counts based on local, componentwise mixing properties.

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.

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

TL;DR

The paper addresses the challenge of proving non-asymptotic variance bounds for Sequential Monte Carlo (SMC) when targeting multimodal distributions where global mixing is inefficient.It develops a framework that bounds variance using local mixing dynamics by decomposing the target measure into mixture components and imposing a generator-decomposition and log-Sobolev constraints, plus a density-ratio bound across levels.Key contributions include a detailed non-asymptotic variance bound, high-probability and total-variation guarantees, and verifiable applicability to Langevin diffusion, Glauber dynamics, Metropolis-Hastings, and TRE-based sampling approaches.The results provide theoretical justification for SMC’s empirical effectiveness in multimodal settings and offer guidance on choosing sampling times and particle counts based on local, componentwise mixing properties.

Abstract

We prove bounds on the variance of a function 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 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.
Paper Structure (19 sections, 24 theorems, 95 equations, 1 algorithm)

This paper contains 19 sections, 24 theorems, 95 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Overview of Sequential Monte Carlo
  3. Notation
  4. Organization of paper
  5. Main Result
  6. Applications
  7. Langevin Diffusion
  8. Gaussian Convolutions and Noise Contrastive Estimation
  9. TV-bounds for Sampling from Telescoping Ratio Estimation
  10. Glauber dynamics
  11. Metropolis-Hastings chain
  12. Overview of Proof
  13. Non-Asymptotic Variance Bounds
  14. Log-Sobolev Bounds & Local Smoothing
  15. Bounding the intra-mode variance
  16. ...and 4 more sections

Key Result

Theorem 2

Suppose Assumption a holds. Then for all $\epsilon > 0$ with $0<\alpha < \frac{1}{\gamma^{2p-2}}$, $\beta = 1 + \sum_i \frac{1}{w_i} > 1$, $p=2^k$ , $k \in \mathbb{N}_{>1}$, choosing yields

Theorems & Definitions (45)

  • Theorem 2: Variance Bound
  • proof
  • Corollary 3: High Probability Bound
  • proof
  • Theorem 4: Total Variation bound
  • proof
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • ...and 35 more