Smoothing unadjusted Langevin algorithms for nonsmooth composite potential functions

This paper addresses a gradient-based Markov Chain Monte Carlo (MCMC) method to sample from the posterior distribution of problems with nonsmooth potential functions. Following the Bayesian paradigm, our potential function will be some of two convex functions, where one of which is smooth. We first approximate the potential function by the so-called forward-backward envelope function, which is a real-valued smooth function with the same critical points as the original one. Then, we incorporate this smoothing technique with the unadjusted Langevin algorithm (ULA), leading to smoothing ULA, called SULA. We next establish non-asymptotic convergence results of SULA under mild assumption on the original potential function. We finally report some numerical results to establish the promising performance of SULA on both synthetic and real chemoinformatics data.

Automated summary

This paper proposes a gradient-based Markov Chain Monte Carlo (MCMC) method for sampling from the posterior distribution of problems with nonsmooth potential functions. By using smoothing techniques, the original potential function is approximated by a smooth function with the same critical points, leading to a smoothing ULA method called SULA. Non-asymptotic convergence results of SULA are established under mild assumptions on the original potential function. Numerical results demonstrate the promising performance of SULA on both synthetic and real chemoinformatics data.