Mixing Time Bounds for the Gibbs Sampler under Isoperimetry

TL;DR

This paper derives new bounds on the mixing times of Gibbs samplers under isoperimetric conditions, extending convergence guarantees beyond log-concave distributions using novel inequalities.

Contribution

It introduces isoperimetric inequalities and coupling techniques to analyze Gibbs sampler convergence for broader classes of distributions.

Findings

Bounds on conductance for Gibbs samplers under Poincaré and log-Sobolev inequalities

Mixing time guarantees extend beyond log-concave distributions

Results valid for log-Lipschitz and log-smooth target distributions

Abstract

We establish bounds on the conductance for the systematic-scan and random-scan Gibbs samplers when the target distribution satisfies a Poincar\'e or log-Sobolev inequality and possesses sufficiently regular conditional distributions. These bounds lead to mixing time guarantees that extend beyond the log-concave setting, offering new insights into the convergence behavior of Gibbs sampling in broader regimes. Moreover, we demonstrate that our results remain valid for log-Lipschitz and log-smooth target distributions. Our approach relies on novel isoperimetric inequalities and a sequential coupling argument for the Gibbs sampler.