AutoStep: Locally adaptive involutive MCMC

TL;DR

AutoStep MCMC introduces a locally adaptive step size mechanism for involutive MCMC kernels, improving robustness and efficiency in sampling complex, multiscale distributions by tuning step sizes based on local geometry.

Contribution

The paper proposes AutoStep MCMC, a novel involutive MCMC method that adaptively selects step sizes at each iteration according to local target geometry, with theoretical guarantees and empirical validation.

Findings

AutoStep MCMC is $ au$-invariant, irreducible, and aperiodic.

It achieves competitive effective sample sizes per unit cost.

The method demonstrates robustness across challenging distributions.

Abstract

Many common Markov chain Monte Carlo (MCMC) kernels can be formulated using a deterministic involutive proposal with a step size parameter. Selecting an appropriate step size is often a challenging task in practice; and for complex multiscale targets, there may not be one choice of step size that works well globally. In this work, we address this problem with a novel class of involutive MCMC methods -- AutoStep MCMC -- that selects an appropriate step size at each iteration adapted to the local geometry of the target distribution. We prove that under mild conditions AutoStep MCMC is $\pi$-invariant, irreducible, and aperiodic, and obtain bounds on expected energy jump distance and cost per iteration. Empirical results examine the robustness and efficacy of our proposed step size selection procedure, and show that AutoStep MCMC is competitive with state-of-the-art methods in terms of effective sample size per unit cost on a range of challenging target distributions.