Stochastic Proximal Langevin Algorithm: Potential Splitting and Nonasymptotic Rates

TL;DR

This paper introduces the Stochastic Proximal Langevin Algorithm (SPLA), a novel sampling method for log-concave distributions that handles complex potentials with stochastic smooth and nonsmooth components, providing theoretical convergence guarantees.

Contribution

The paper presents SPLA, a new algorithm that generalizes Langevin sampling to potentials with stochastic smooth and nonsmooth parts, with proven nonasymptotic convergence rates.

Findings

Achieves nonasymptotic sublinear convergence under convexity.

Attains linear convergence under strong convexity.

Demonstrates efficiency through Bayesian learning simulations.

Abstract

We propose a new algorithm---Stochastic Proximal Langevin Algorithm (SPLA)---for sampling from a log concave distribution. Our method is a generalization of the Langevin algorithm to potentials expressed as the sum of one stochastic smooth term and multiple stochastic nonsmooth terms. In each iteration, our splitting technique only requires access to a stochastic gradient of the smooth term and a stochastic proximal operator for each of the nonsmooth terms. We establish nonasymptotic sublinear and linear convergence rates under convexity and strong convexity of the smooth term, respectively, expressed in terms of the KL divergence and Wasserstein distance. We illustrate the efficiency of our sampling technique through numerical simulations on a Bayesian learning task.