Exponential ergodicity of mirror-Langevin diffusions

Abstract

Motivated by the problem of sampling from ill-conditioned log-concave distributions, we give a clean non-asymptotic convergence analysis of mirror-Langevin diffusions as introduced in Zhang et al. (2020). As a special case of this framework, we propose a class of diffusions called Newton-Langevin diffusions and prove that they converge to stationarity exponentially fast with a rate which not only is dimension-free, but also has no dependence on the target distribution. We give an application of this result to the problem of sampling from the uniform distribution on a convex body using a strategy inspired by interior-point methods. Our general approach follows the recent trend of linking sampling and optimization and highlights the role of the chi-squared divergence. In particular, it yields new results on the convergence of the vanilla Langevin diffusion in Wasserstein distance.

27 pages, 10 figures

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• Convergence of Stein Variational Gradient Descent under a Weaker Smoothness Condition
• Sampling with Mirrored Stein Operators
• Sampling From the Wasserstein Barycenter
• Penalized Langevin dynamics with vanishing penalty for smooth and log-concave targets
• Efficient Multimodal Sampling via Tempered Distribution Flow
• Langevin Monte Carlo for strongly log-concave distributions: Randomized midpoint revisited