Improving the convergence of reversible samplers

TL;DR

This paper develops general principles for designing reversible and irreversible perturbations to Markov processes, improving convergence rates in Monte Carlo sampling across various processes like chains and diffusions.

Contribution

It introduces a unified framework for enhancing Markov sampler efficiency through perturbations, applicable to multiple process types and performance criteria.

Findings

Perturbations improve spectral gap and asymptotic variance.

Specific constructions provided for chains and diffusions.

Large deviation functionals used to measure performance improvements.

Abstract

In Monte-Carlo methods the Markov processes used to sample a given target distribution usually satisfy detailed balance, i.e. they are time-reversible. However, relatively recent results have demonstrated that appropriate reversible and irreversible perturbations can accelerate convergence to equilibrium. In this paper we present some general design principles which apply to general Markov processes. Working with the generator of Markov processes, we prove that for some of the most commonly used performance criteria, i.e., spectral gap, asymptotic variance and large deviation functionals, sampling is improved for appropriate reversible and irreversible perturbations of some initially given reversible sampler. Moreover we provide specific constructions for such reversible and irreversible perturbations for various commonly used Markov processes, such as Markov chains and diffusions. In the case of diffusions, we make the discussion more specific using the large deviations rate function as a measure of performance.