Primal Dual Interpretation of the Proximal Stochastic Gradient Langevin Algorithm

TL;DR

This paper introduces a primal-dual framework for analyzing the Proximal Stochastic Gradient Langevin Algorithm (PSGLA) in sampling from log-concave distributions, establishing complexity bounds and extending applicability to non-supported cases.

Contribution

It provides a strong duality result for the sampling problem and interprets PSGLA as a primal-dual algorithm, offering new complexity bounds especially when the potential is strongly convex.

Findings

PSGLA has complexity O(1/ε^2) in Wasserstein distance for strongly convex potentials.

The complexity of Projected Langevin Algorithm is O(1/ε^{12}) in total variation for convex potentials.

The duality approach extends analysis to cases where the target distribution is not fully supported.

Abstract

We consider the task of sampling with respect to a log concave probability distribution. The potential of the target distribution is assumed to be composite, \textit{i.e.}, written as the sum of a smooth convex term, and a nonsmooth convex term possibly taking infinite values. The target distribution can be seen as a minimizer of the Kullback-Leibler divergence defined on the Wasserstein space (\textit{i.e.}, the space of probability measures). In the first part of this paper, we establish a strong duality result for this minimization problem. In the second part of this paper, we use the duality gap arising from the first part to study the complexity of the Proximal Stochastic Gradient Langevin Algorithm (PSGLA), which can be seen as a generalization of the Projected Langevin Algorithm. Our approach relies on viewing PSGLA as a primal dual algorithm and covers many cases where the target distribution is not fully supported. In particular, we show that if the potential is strongly convex, the complexity of PSGLA is $O(1/\varepsilon^2)$ in terms of the 2-Wasserstein distance. In contrast, the complexity of the Projected Langevin Algorithm is $O(1/\varepsilon^{12})$ in terms of total variation when the potential is convex.