Exponential ergodicity of mirror-Langevin diffusions

Part of Advances in Neural Information Processing Systems 33 pre-proceedings (NeurIPS 2020)

Bibtex »Paper »Supplemental »

Bibtek download is not availble in the pre-proceeding


Sinho Chewi, Thibaut Le Gouic, Chen Lu, Tyler Maunu, Philippe Rigollet, Austin Stromme


<p>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.</p>