The Performance Of The Unadjusted Langevin Algorithm Without Smoothness Assumptions

TL;DR

This paper introduces a Langevin-based sampling algorithm that works effectively without smoothness assumptions, using a geometric one-sided Lipschitz condition, and provides non-asymptotic convergence guarantees.

Contribution

It presents a simple Langevin algorithm that operates under weaker regularity conditions and offers non-asymptotic convergence guarantees for sampling and optimization.

Findings

Effective sampling without smoothness assumptions

Non-asymptotic convergence guarantees in Wasserstein distance

Applicable to discontinuous log-gradients

Abstract

In this article, we study the problem of sampling from distributions whose densities are not necessarily smooth nor logconcave. We propose a simple Langevin-based algorithm that does not rely on popular but computationally challenging techniques, such as the Moreau-Yosida envelope or Gaussian smoothing, and show consequently that the performance of samplers like ULA does not necessarily degenerate arbitrarily with low regularity. In particular, we show that the Lipschitz or H\"older continuity assumption can be replaced by a geometric one-sided Lipschitz condition that allows even for discontinuous log-gradients. We derive non-asymptotic guarantees for the convergence of the algorithm to the target distribution in Wasserstein distances. Non-asymptotic bounds are also provided for the performance of the algorithm as an optimizer, specifically for the solution of associated excess risk optimization problems.