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

2026-01-29

AI Summary

This work addresses the lack of explicit spectral gap bounds for Metropolis algorithms targeting non-smooth probability distributions—a gap that has hindered both theoretical analysis and practical applications. For the first time, we establish explicit spectral gap bounds for both the Random Walk Metropolis and the Metropolis-adjusted Langevin algorithms under non-smooth target densities, thereby relaxing the commonly imposed strong log-concavity assumption. Our results extend to a broader class of distributions satisfying either a Poincaré or a logarithmic Sobolev inequality. The analysis integrates isoperimetric inequalities, spectral theory of Markov operators, and recent advances in functional inequalities. The derived bounds are not only theoretically rigorous but also validated through numerical experiments, significantly broadening the theoretical foundation and applicability of Metropolis-type algorithms.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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 discuss 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.

Problem

• spectral gap
• Metropolis algorithms
• non-smooth distributions
• isoperimetry
• convergence rate

Innovation

• spectral gap
• non-smooth distributions
• Metropolis algorithms
• isoperimetry
• log-Sobolev inequality