Variance reduction for irreversible Langevin samplers and diffusion on graphs

TL;DR

This paper investigates how increasing irreversible drift in Langevin samplers reduces variance and enhances sampling efficiency, especially when the process decomposes into slow and fast components on a graph-like structure.

Contribution

It characterizes the asymptotic variance behavior of irreversible Langevin samplers with large drift and links variance reduction to the process's decomposition into slow and fast motions.

Findings

Asymptotic variance decreases monotonically with drift magnitude.

Large irreversible drift induces a process decomposition into slow and fast components.

Limiting variance corresponds to a diffusion process on a graph.

Abstract

In recent papers it has been demonstrated that sampling a Gibbs distribution from an appropriate time-irreversible Langevin process is, from several points of view, advantageous when compared to sampling from a time-reversible one. Adding an appropriate irreversible drift to the overdamped Langevin equation results in a larger large deviations rate function for the empirical measure of the process, a smaller variance for the long time average of observables of the process, as well as a larger spectral gap. In this work, we concentrate on irreversible Langevin samplers with a drift of increasing intensity. The asymptotic variance is monotonically decreasing with respect to the growth of the drift and we characterize its limiting behavior. For a Gibbs measure whose potential has one or more critical points, adding a large irreversible drift results in a decomposition of the process in a slow and fast component with fast motion along the level sets of the potential and slow motion in the orthogonal direction. This result helps understanding the variance reduction, which can be explained at the process level by the induced fast motion of the process along the level sets of the potential. The limit of the asymptotic variance as the magnitude of the irreversible perturbation grows is the asymptotic variance associated to the limiting slow motion. The latter is a diffusion process on a graph.