Spectral Gap of Metropolis Algorithms for Non-smooth Distributions under Isoperimetry

TL;DR

This paper provides explicit spectral gap bounds for Metropolis algorithms applied to non-smooth distributions, extending previous results to broader classes of target distributions under isoperimetric conditions.

Contribution

It derives the first explicit spectral gap bounds for Metropolis algorithms on non-smooth targets and extends these bounds to distributions satisfying Poincare or log-Sobolev inequalities.

Findings

Explicit spectral gap bounds for non-smooth distributions.

Extension of bounds to targets with Poincare or log-Sobolev inequalities.

Numerical experiments supporting theoretical results.

Abstract

Metropolis algorithms are classical tools for sampling from target distributions, with broad applications in statistics and scientific computing. Their convergence speed is governed by the spectral gap of the associated Markov operator. Recently, Andrieu et al. (2024) derived the first explicit bounds for the spectral gap of Random--Walk Metropolis when the target distribution is smooth and strongly log-concave. However, existing literature rarely discusses non-smooth targets. In this work, we derive explicit spectral gap bounds for the random-walk Metropolis and Metropolis--adjusted Langevin algorithms over a broad class of non-smooth distributions. Moreover, combining our analysis with a recent result in Goyal et al. (2025), we extend these bounds to targets satisfying a Poincare or log-Sobolev inequality, beyond the strongly log-concave setting. Our theoretical results are further supported by numerical experiments.