Ergodicity of the infinite swapping algorithm at low temperature

TL;DR

This paper rigorously analyzes the ergodic properties of the infinite-swapping algorithm at low temperatures, demonstrating its effectiveness in reducing energy barriers and improving sampling efficiency compared to classical methods.

Contribution

The paper provides a rigorous analysis of the ergodic properties of the infinite-swapping algorithm, including spectral gap and log-Sobolev constant estimates, showing its advantages over traditional Langevin dynamics.

Findings

Effective energy barrier is significantly reduced by the isa.

Sampling efficiency is exponentially improved with the isa.

The isa outperforms overdamped Langevin dynamics in simulated annealing.

Abstract

Sampling Gibbs measures at low temperatures is an important task but computationally challenging. Numerical evidence suggests that the infinite-swapping algorithm (isa) is a promising method. The isa can be seen as an improvement of the replica methods. We rigorously analyze the ergodic properties of the isa in the low temperature regime, deducing an Eyring-Kramers formula for the spectral gap (or Poincar\'e constant) and an estimate for the log-Sobolev constant. Our main results indicate that the effective energy barrier can be reduced drastically using the isa compared to the classical overdamped Langevin dynamics. As a corollary, we derive a deviation inequality showing that sampling is also improved by an exponential factor. Finally, we study simulated annealing for the isa and prove that the isa again outperforms the overdamped Langevin dynamics.