Bregman Proximal Langevin Monte Carlo via Bregman--Moreau Envelopes
This work addresses the challenge of efficient sampling for nonsmooth distributions in machine learning and optimization, offering incremental improvements over existing Langevin Monte Carlo techniques.
The authors tackled the problem of sampling from nonsmooth convex composite distributions by proposing Langevin Monte Carlo algorithms that use Bregman--Moreau envelopes and Bregman proximity operators, extending existing methods to handle nonsmooth potentials and demonstrating improved performance in tasks where prior methods fail.
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.