Analysis of Langevin Monte Carlo via convex optimization

TL;DR

This paper reinterprets the Unadjusted Langevin Algorithm as a convex optimization method on Wasserstein space, providing non-asymptotic analysis and proposing new algorithms for non-smooth distributions suitable for large-scale Bayesian inference.

Contribution

It introduces a novel convex optimization perspective on Langevin algorithms and develops two new sampling methods for non-smooth distributions, extending SGLD.

Findings

Non-asymptotic convergence analysis of Langevin methods

Development of two new algorithms for non-smooth targets

Extension of SGLD for large-scale Bayesian inference

Abstract

In this paper, we provide new insights on the Unadjusted Langevin Algorithm. We show that this method can be formulated as a first order optimization algorithm of an objective functional defined on the Wasserstein space of order $2$. Using this interpretation and techniques borrowed from convex optimization, we give a non-asymptotic analysis of this method to sample from logconcave smooth target distribution on $\mathbb{R}^d$. Based on this interpretation, we propose two new methods for sampling from a non-smooth target distribution, which we analyze as well. Besides, these new algorithms are natural extensions of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm, which is a popular extension of the Unadjusted Langevin Algorithm. Similar to SGLD, they only rely on approximations of the gradient of the target log density and can be used for large-scale Bayesian inference.