Unbiased Estimation using the Underdamped Langevin Dynamics

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 to admit the probability of interest as a stationary measure. We develop a novel scheme based upon doubly randomized estimation, which requires access only to time-discretized versions of the dynamics and are the ones that are used in practical algorithms. 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 that verify our theory, which include challenging examples from Bayesian statistics and statistical physics.