Rapid convergence of tempering chains to multimodal Gibbs measures

TL;DR

This paper proves polynomial lower bounds on the spectral gaps of tempering chains for multimodal Gibbs measures, ensuring rapid convergence without detailed landscape knowledge.

Contribution

It introduces a novel Lyapunov function approach to analyze spectral gaps of tempering chains for broad classes of potentials.

Findings

Spectral gaps have polynomial lower bounds of order 11 and 12.

Analysis applies to a broad class of potentials beyond mixture models.

Method does not require explicit structural information of the energy landscape.

Abstract

We study the spectral gaps of parallel and simulated tempering chains targeting multimodal Gibbs measures. In particular, we consider chains constructed from Metropolis random walks that preserve the Gibbs distributions at a sequence of harmonically spaced temperatures. We prove that their spectral gaps admit polynomial lower bounds of order $11$ and $12$ in terms of the low target temperature. The analysis applies to a broad class of potentials, beyond mixture models, without requiring explicit structural information on the energy landscape. The main idea is to decompose the state space and construct a Lyapunov function based on a suitably perturbed potential, which allows us to establish lower bounds on the local spectral gaps.