Variance reduction for irreversible Langevin samplers and diffusion on graphs

Luc Rey-Bellet,Konstantinos Spiliopoulos

doi:10.1214/ecp.v20-3855

Luc Rey-Bellet, Konstantinos Spiliopoulos

Open Access

https://doi.org/10.1214/ecp.v20-3855

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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. Correspondingly the limit of the asymptotic variance is the asymptotic variance of the limiting slow motion which is a diffusion process on a graph.

Highlights

It is often the case that one is given a high dimensional distribution π(dx) which is known only up to normalizing constants, a state space E and an observable f and the goal is to compute an integral of the form f = E f (x)π(dx)
In [13], this criterion is used as a guide to design and analyze non-reversible Markov processes and compare them with reversible ones and we prove that the large deviations rate function monotonically increases under the addition of an irreversible drift
We find that the asymptotic variance of the estimator is monotonically decreasing in δ = 1/

Summary

Introduction

There, the authors studied the behavior of the spectral gap for diffusions on compact manifolds with U = 0 and a one-parameter families of perturbations 1 C for some divergence free vector field C In those papers the behavior of the spectral gap is related to the ergodic properties of the flow generated by C (for example if the flow is weak-mixing the second largest eigenvalue tends to 0 as → 0). The fast motion on constant potential surfaces decreases the variance as the phase space is explored faster and the limit of the asymptotic variance as → 0 is the asymptotic variance of a one-dimensional estimation problem on a graph, which is where, in the limit, the slow component of the process lives.

Statement of the problem and main results

The averaging problem

Limiting behavior of the asymptotic variance

Numerical Simulations