A Proximal Algorithm for Sampling from Non-convex Potentials

TL;DR

This paper introduces a proximal sampling algorithm for non-convex, semi-smooth potentials that extends previous methods, achieving better complexity in sampling from challenging distributions satisfying certain inequalities.

Contribution

It develops a practical proximal sampling algorithm based on rejection sampling and the alternating sampling framework for non-convex, semi-smooth potentials, extending prior work.

Findings

Achieves better complexity than existing methods in most cases.

Extends recent algorithms to non-convex, semi-smooth settings.

Provides a practical realization of the ASF for challenging sampling tasks.

Abstract

We study sampling problems associated with non-convex potentials that meanwhile lack smoothness. In particular, we consider target distributions that satisfy either logarithmic-Sobolev inequality or Poincar\'e inequality. Rather than smooth, the potentials are assumed to be semi-smooth or the summation of multiple semi-smooth functions. We develop a sampling algorithm that resembles proximal algorithms in optimization for this challenging sampling task. Our algorithm is based on a special case of Gibbs sampling known as the alternating sampling framework (ASF). The key contribution of this work is a practical realization of the ASF based on rejection sampling in the non-convex and semi-smooth setting. This work extends the recent algorithm in \cite{LiaChe21,LiaChe22} for non-smooth/semi-smooth log-concave distribution to the setting with non-convex potentials. In almost all the cases of sampling considered in this work, our proximal sampling algorithm achieves better complexity than all existing methods.