Curvature, concentration and error estimates for Markov chain Monte Carlo

TL;DR

This paper derives explicit, nonasymptotic convergence and deviation estimates for Markov chain empirical means under a positive curvature condition, extending geometric ergodicity concepts.

Contribution

It introduces a curvature-based framework to obtain explicit convergence rates and deviation bounds for Markov chains, generalizing Ricci curvature ideas.

Findings

Explicit convergence rate estimates under positive curvature

Gaussian and exponential deviation controls established

Generalization of Ricci curvature to Markov chain analysis

Abstract

We provide explicit nonasymptotic estimates for the rate of convergence of empirical means of Markov chains, together with a Gaussian or exponential control on the deviations of empirical means. These estimates hold under a "positive curvature" assumption expressing a kind of metric ergodicity, which generalizes the Ricci curvature from differential geometry and, on finite graphs, amounts to contraction under path coupling.