The Langevin Monte Carlo algorithm in the non-smooth log-concave case

Abstract

We prove non asymptotic polynomial bounds on the convergence of the Langevin Monte Carlo algorithm in the case where the potential is a convex function which is globally Lipschitz on its domain, typically the maximum of a finite number of affine functions on an arbitrary convex set. In particular the potential is not assumed to be gradient Lipschitz, in contrast with most existing works on the topic.

v3: added some explanations about the existence of the diffusion and removed the last section which was redundant with arXiv:1704.04752