The Forward-Backward Envelope for Sampling with the Overdamped Langevin Algorithm

TL;DR

This paper introduces a forward-backward envelope method for sampling from non-smooth distributions using Langevin dynamics, improving upon existing algorithms by preserving the MAP estimator.

Contribution

It proposes a novel forward-backward envelope approach for Langevin-based sampling that handles non-smooth densities while maintaining the MAP estimate.

Findings

The method effectively samples from non-smooth distributions.

Numerical experiments validate the theoretical advantages.

The approach outperforms traditional algorithms like MYULA.

Abstract

In this paper, we analyse a proximal method based on the idea of forward-backward splitting for sampling from distributions with densities that are not necessarily smooth. In particular, we study the non-asymptotic properties of the Euler-Maruyama discretization of the Langevin equation, where the forward-backward envelope is used to deal with the non-smooth part of the dynamics. An advantage of this envelope, when compared to widely-used Moreu-Yoshida one and the MYULA algorithm, is that it maintains the MAP estimator of the original non-smooth distribution. We also study a number of numerical experiments that corroborate that support our theoretical findings.