Unbiased Estimation using Underdamped Langevin Dynamics

TL;DR

This paper introduces a novel unbiased estimation method using underdamped Langevin dynamics, effectively removing discretization bias and finite iteration bias, with proven finite variance and demonstrated success in complex Bayesian and physics applications.

Contribution

It develops a new doubly randomized scheme for unbiased estimation with underdamped Langevin dynamics, addressing discretization and iteration biases.

Findings

Estimator has finite variance under standard assumptions.

Method achieves finite expected cost or high-probability finite cost.

Numerical experiments confirm theoretical properties in complex scenarios.

Abstract

In this work we consider the unbiased estimation of expectations w.r.t.~probability measures that have non-negative Lebesgue density, and which are known point-wise up-to a normalizing constant. We focus upon developing an unbiased method via the underdamped Langevin dynamics, which has proven to be popular of late due to applications in statistics and machine learning. Specifically in continuous-time, the dynamics can be constructed {so that as the time goes to infinity they} admit the probability of interest as a stationary measure. {In many cases, time-discretized versions of the underdamped Langevin dynamics are used in practice which are run only with a fixed number of iterations.} We develop a novel scheme based upon doubly randomized estimation as in \cite{ub_grad,disc_model}, which requires access only to time-discretized versions of the dynamics. {The proposed scheme aims to remove the dicretization bias and the bias resulting from running the dynamics for a finite number of iterations}. We prove, under standard assumptions, that our estimator is of finite variance and either has finite expected cost, or has finite cost with a high probability. To illustrate our theoretical findings we provide numerical experiments which verify our theory, which include challenging examples from Bayesian statistics and statistical physics.