Quantitative contraction rates for Markov chains on general state spaces

TL;DR

This paper develops quantitative lower bounds on contraction rates for Markov chains on general state spaces, extending techniques from diffusion processes to analyze convergence properties of various Markov algorithms.

Contribution

It introduces a method to derive explicit contraction bounds for Markov chains, especially those dominated by small local moves, applicable to high-dimensional sampling algorithms.

Findings

Derived lower bounds for contraction rates of Markov chains.

Applied bounds to Euler discretizations of SDEs with non-globally contractive drifts.

Analyzed the Metropolis adjusted Langevin algorithm in high-dimensional, non-log-concave settings.

Abstract

We investigate the problem of quantifying contraction coefficients of Markov transition kernels in Kantorovich ($L^1$ Wasserstein) distances. For diffusion processes, relatively precise quantitative bounds on contraction rates have recently been derived by combining appropriate couplings with carefully designed Kantorovich distances. In this paper, we partially carry over this approach from diffusions to Markov chains. We derive quantitative lower bounds on contraction rates for Markov chains on general state spaces that are powerful if the dynamics is dominated by small local moves. For Markov chains on $\mathbb{R^d}$ with isotropic transition kernels, the general bounds can be used efficiently together with a coupling that combines maximal and reflection coupling. The results are applied to Euler discretizations of stochastic differential equations with non-globally contractive drifts, and to the Metropolis adjusted Langevin algorithm for sampling from a class of probability measures on high dimensional state spaces that are not globally log-concave.