Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem

TL;DR

This paper frames sampling as an optimization problem in the space of measures, analyzing Langevin dynamics and proposing the symmetrized Langevin algorithm to reduce bias in sampling, especially for Gaussian measures.

Contribution

It introduces the symmetrized Langevin algorithm (SLA) as a novel method to reduce bias in measure-space sampling, with theoretical and practical advantages over ULA.

Findings

SLA has smaller bias than ULA for Gaussian measures

SLA is consistent for Gaussian target measures

Various algorithms like gradient descent and proximal gradient are consistent for Gaussian targets

Abstract

We study sampling as optimization in the space of measures. We focus on gradient flow-based optimization with the Langevin dynamics as a case study. We investigate the source of the bias of the unadjusted Langevin algorithm (ULA) in discrete time, and consider how to remove or reduce the bias. We point out the difficulty is that the heat flow is exactly solvable, but neither its forward nor backward method is implementable in general, except for Gaussian data. We propose the symmetrized Langevin algorithm (SLA), which should have a smaller bias than ULA, at the price of implementing a proximal gradient step in space. We show SLA is in fact consistent for Gaussian target measure, whereas ULA is not. We also illustrate various algorithms explicitly for Gaussian target measure, including gradient descent, proximal gradient, and Forward-Backward, and show they are all consistent.