Bregman Proximal Langevin Monte Carlo via Bregman--Moreau Envelopes

TL;DR

This paper introduces novel Langevin Monte Carlo algorithms that efficiently sample from nonsmooth convex composite distributions by leveraging Bregman divergences and Moreau envelopes, extending existing methods to nonsmooth settings.

Contribution

The paper develops Langevin Monte Carlo algorithms using Bregman--Moreau envelopes, enabling efficient sampling from nonsmooth distributions and generalizing mirror descent techniques.

Findings

Algorithms outperform existing methods on challenging sampling tasks.

Effective in handling nonsmooth convex potentials.

Demonstrated improved convergence in practical experiments.

Abstract

We propose efficient Langevin Monte Carlo algorithms for sampling distributions with nonsmooth convex composite potentials, which is the sum of a continuously differentiable function and a possibly nonsmooth function. We devise such algorithms leveraging recent advances in convex analysis and optimization methods involving Bregman divergences, namely the Bregman--Moreau envelopes and the Bregman proximity operators, and in the Langevin Monte Carlo algorithms reminiscent of mirror descent. The proposed algorithms extend existing Langevin Monte Carlo algorithms in two aspects -- the ability to sample nonsmooth distributions with mirror descent-like algorithms, and the use of the more general Bregman--Moreau envelope in place of the Moreau envelope as a smooth approximation of the nonsmooth part of the potential. A particular case of the proposed scheme is reminiscent of the Bregman proximal gradient algorithm. The efficiency of the proposed methodology is illustrated with various sampling tasks at which existing Langevin Monte Carlo methods are known to perform poorly.