# Local and Global Contraction Principles for MCMC Mixing

arXiv: https://arxiv.org/abs/2606.03033 · HTML: https://mlsanity.com/paper/2606.03033/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 52 references that are in the map.
- First posted on arXiv 2 Jun 2026.

## Builds on (most cited first, 28 of 28)

- [Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis](https://mlsanity.com/paper/1702.03849/) (2017)
- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016)
- [Analysis of Langevin Monte Carlo via convex optimization](https://mlsanity.com/paper/1802.09188/) (2018)
- [Log-concave sampling: Metropolis-Hastings algorithms are fast](https://mlsanity.com/paper/1801.02309/) (2018)
- [User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient](https://mlsanity.com/paper/1710.00095/) (2017)
- [Privacy Amplification by Iteration](https://mlsanity.com/paper/1808.06651/) (2018)
- [Convergence of Langevin MCMC in KL-divergence](https://mlsanity.com/paper/1705.09048/) (2017)
- [Sharp convergence rates for Langevin dynamics in the nonconvex setting](https://mlsanity.com/paper/1805.01648/) (2018)
- [Sampling Can Be Faster Than Optimization](https://mlsanity.com/paper/1811.08413/) (2018)
- [Privacy of Noisy Stochastic Gradient Descent: More Iterations without More Privacy Loss](https://mlsanity.com/paper/2205.13710/) (2022)
- [Optimal dimension dependence of the Metropolis-Adjusted Langevin Algorithm](https://mlsanity.com/paper/2012.12810/) (2020)
- [On the Convergence of Langevin Monte Carlo: The Interplay between Tail Growth and Smoothness](https://mlsanity.com/paper/2005.13097/) (2020)
- [Sampling from a log-concave distribution with Projected Langevin Monte Carlo](https://mlsanity.com/paper/1507.02564/) (2015)
- [Convergence of Langevin Monte Carlo in Chi-Squared and Renyi Divergence](https://mlsanity.com/paper/2007.11612/) (2020)
- [Non-asymptotic bounds for sampling algorithms without log-concavity](https://mlsanity.com/paper/1808.07105/) (2018)
- [Projected Stochastic Gradient Langevin Algorithms for Constrained Sampling and Non-Convex Learning](https://mlsanity.com/paper/2012.12137/) (2020)
- [Towards a Complete Analysis of Langevin Monte Carlo: Beyond Poincar'e Inequality](https://mlsanity.com/paper/2303.03589/) (2023)
- [Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Sampling](https://mlsanity.com/paper/2210.08448/) (2022)
- [Privacy Amplification by Mixing and Diffusion Mechanisms](https://mlsanity.com/paper/1905.12264/) (2019)
- [Bounding the error of discretized Langevin algorithms for non-strongly log-concave targets](https://mlsanity.com/paper/1906.08530/) (2019)
- [Nonconvex sampling with the Metropolis-adjusted Langevin algorithm](https://mlsanity.com/paper/1902.08452/) (2019)
- [Fast Convergence of Phi-Divergence Along the Unadjusted Langevin Algorithm and Proximal Sampler](https://mlsanity.com/paper/2410.10699/) (2024)
- [Privacy Loss of Noisy Stochastic Gradient Descent Might Converge Even for Non-Convex Losses](https://mlsanity.com/paper/2305.09903/) (2023)
- [Comparison of Markov chains via weak Poincar'e inequalities with application to pseudo-marginal MCMC](https://mlsanity.com/paper/2112.05605/) (2021)
- [Fast Conditional Mixing of MCMC Algorithms for Non-log-concave Distributions](https://mlsanity.com/paper/2306.10506/) (2023)
- [On importance sampling and independent Metropolis-Hastings with an unbounded weight function](https://mlsanity.com/paper/2411.09514/) (2024)

## Abstract (arXiv)

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 Eγ-divergence with γ≥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 Eγ-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, chi2, and R'enyi 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 CR=w≤ 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 HR=π(w>R), recovering sharp existing moment-based convergence rates when mathbb Eq[wp]<∞ for some p>1, while remaining effective in heavy-tailed regimes where no finite moment of order p>1 exists.