Optimal friction matrix for underdamped Langevin sampling

TL;DR

This paper introduces a systematic method to optimize the friction coefficient in underdamped Langevin dynamics, improving sampling efficiency and reducing variance in Bayesian inference tasks.

Contribution

It provides a gradient-based optimization procedure for the friction coefficient using solutions to a Poisson equation, with practical algorithms demonstrated on Bayesian posterior estimation.

Findings

Reduced variance in Bayesian posterior mean estimation

Improved sampling efficiency over traditional Langevin methods

Effective in both full and stochastic gradient scenarios

Abstract

A systematic procedure for optimising the friction coefficient in underdamped Langevin dynamics as a sampling tool is given by taking the gradient of the associated asymptotic variance with respect to friction. We give an expression for this gradient in terms of the solution to an appropriate Poisson equation and show that it can be approximated by short simulations of the associated first variation/tangent process under concavity assumptions on the log density. Our algorithm is applied to the estimation of posterior means in Bayesian inference problems and reduced variance is demonstrated when compared to the original underdamped and overdamped Langevin dynamics in both full and stochastic gradient cases.