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

TL;DR

This paper establishes non-asymptotic convergence bounds for the Langevin Monte Carlo algorithm applied to non-smooth, convex, Lipschitz potentials, expanding its applicability beyond smooth cases.

Contribution

It provides the first polynomial convergence bounds for Langevin Monte Carlo with non-smooth convex potentials, relaxing the gradient Lipschitz assumption.

Findings

Polynomial convergence bounds are proven for non-smooth convex potentials.

The results apply to potentials that are maxima of affine functions on convex sets.

This work broadens the theoretical understanding of Langevin Monte Carlo in non-smooth settings.

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.