Local and Global Contraction Principles for MCMC Mixing

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 ${\mathsf{E}}_{\hat{I}}^3$-divergence with ${\hat{I}}^3\ge 1$. For projected Langevin Monte Carlo on a compact convex domain, we show that Gaussian smoothing yields an explicit global contraction coefficient for the ${\mathsf{E}}_{\hat{I}}^3$-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, $\ddot{I}{‡}^2$, and Rényi divergences. For independent Metropolis--Hastings with target $\ddot{I}€$, proposal $q$, and unbounded importance weight $w=d\ddot{I}€/dq$, 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 $H_R=\ddot{I}€(w>R)$, 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.