{"ID":3006179,"CreatedAt":"2026-06-03T03:09:48.883664427Z","UpdatedAt":"2026-06-04T19:14:31.964469513Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.03033","arxiv_id":"2606.03033","title":"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 $\\mathsf E_γ$-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 $H_R=π(w\u003eR)$, recovering sharp existing moment-based convergence rates when $\\mathbb E_q[w^p]\u003c\\infty$ for some $p\u003e1$, while remaining effective in heavy-tailed regimes where no finite moment of order $p\u003e1$ exists.","short_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 sh...","url_abs":"https://arxiv.org/abs/2606.03033","url_pdf":"https://arxiv.org/pdf/2606.03033v1","authors":"[\"Alireza Daeijavad\",\"Shahab Asoodeh\"]","published":"2026-06-02T02:16:52Z","proceeding":"cs.IT","tasks":"[\"cs.IT\",\"math.ST\",\"stat.CO\",\"stat.ML\"]","methods":"[]","has_code":false}