Restricted Spectral Gap Decomposition for Simulated Tempering Targeting Mixture Distributions

TL;DR

This paper introduces a new theoretical framework for analyzing simulated tempering algorithms, providing bounds on their efficiency when sampling from complex mixture distributions, especially in high-dimensional settings.

Contribution

The paper presents a novel decomposition theorem that bounds the restricted spectral gap of simulated tempering with local MCMC, extending applicability to broader scenarios.

Findings

Spectral gap bounds scale polynomially with mode separation

Complexity scales logarithmically with inverse accuracy

Exponential dependence on the dimension d

Abstract

Simulated tempering is a widely used strategy for sampling from multimodal distributions. In this paper, we consider simulated tempering combined with an arbitrary local Markov chain Monte Carlo sampler and present a new decomposition theorem that provides a lower bound on the restricted spectral gap of the algorithm for sampling from mixture distributions. By working with the restricted spectral gap, the applicability of our results is extended to broader settings such as when the usual spectral gap is difficult to bound or becomes degenerate. We demonstrate the application of our theoretical results by analyzing simulated tempering combined with random walk Metropolis--Hastings for sampling from mixtures of Gaussian distributions. Our complexity bound scales polynomially with the separation between modes, logarithmically with $1/\varepsilon$, where $\varepsilon$ denotes the target accuracy in total variation distance, and exponentially with the dimension $d$.