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.
Cite
@article{arxiv.2005.09669,
title = {Exponential ergodicity of mirror-Langevin diffusions},
author = {Sinho Chewi and Thibaut Le Gouic and Chen Lu and Tyler Maunu and Philippe Rigollet and Austin J. Stromme},
journal= {arXiv preprint arXiv:2005.09669},
year = {2020}
}
Comments
27 pages, 10 figures