Local and Global Contraction Principles for MCMC Mixing

• URL: http://arxiv.org/abs/2606.03033v1
• Date: Tue, 02 Jun 2026 02:16:52 GMT
• Title: Local and Global Contraction Principles for MCMC Mixing
• Abstract summary: We develop a contraction-based framework for proving mixing-time bounds for Monte Carlo algorithms.<n>For Langevin Langevin's compact convex domain, we show an explicit global contraction coefficient.<n>We recover sharp existing moment-based convergence rates when $mathbb E_q[wp]infty$ for somep regimes $p1$ where no finite moment exists.
• Score: 2.889268075288957
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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