Local and Global Contraction Principles for MCMC Mixing

2026-06-01

AI Summary

This work addresses the challenge of analyzing mixing times for Markov chain Monte Carlo (MCMC) algorithms under non-convex potentials and heavy-tailed target distributions by introducing a unified contraction framework based on $\mathsf{E}_\gamma$-divergence. The framework incorporates Gaussian smoothing to derive an explicit global contraction coefficient and introduces a local contraction coefficient to handle unbounded importance weights, thereby effectively accommodating heavy-tailed settings. It applies broadly to algorithms such as projected Langevin Monte Carlo and independent Metropolis–Hastings. For the former, it establishes, for the first time, explicit exponential convergence rates under KL, $\chi^2$, and Rényi divergences; for the latter, it recovers or improves existing convergence bounds under either finite-moment or heavy-tailed conditions.

Abstract

We develop a contraction-based framework for proving mixing-time bounds for Markov chain Monte Carlo algorithms. The framework is built around global and local contraction coefficients of Markov kernels under the $\mathsf E_γ$-divergence with $γ\ge1$. For projected Langevin Monte Carlo on a compact convex domain, we show that Gaussian smoothing yields an explicit global contraction coefficient for the $\mathsf E_γ$-divergence. This gives a direct proof of exponential convergence to the discretized stationary distribution for general smooth, possibly non-convex potentials. The rate is explicit, accommodates arbitrary random-batch sampling schemes, and yields convergence guarantees for several divergences, including KL, $χ^2$, and Rényi divergences. For independent Metropolis--Hastings with target $π$, proposal $q$, and unbounded importance weight $w=dπ/dq$, global contraction coefficients are typically trivial. We therefore introduce a local contraction coefficient on the core $C_R=\{w\le R\}$ and prove that it controls the rejection profile on the core. This yields warm-start convergence bounds governed by the local contraction coefficient and the tail profile $H_R=π(w>R)$, recovering sharp existing moment-based convergence rates when $\mathbb E_q[w^p]<\infty$ for some $p>1$, while remaining effective in heavy-tailed regimes where no finite moment of order $p>1$ exists.

Problem

• MCMC mixing
• contraction coefficients
• non-convex potentials
• heavy-tailed distributions
• convergence guarantees

Innovation

• contraction coefficient
• MCMC mixing
• E_gamma divergence
• Langevin Monte Carlo
• Metropolis–Hastings