Unbiased estimation using underdamped langevin dynamics

In this work we consider the unbiased estimation of expectations w.r.t. probability measures that have nonnegative Lebesgue density and which are known pointwise 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 as in [J. Heng, J. Houssineau, and A. Jasra, On Unbiased Score Estimation for Partially Observed Diffusions, preprint, 2021] and [J. Heng, A. Jasra, K. J. H. Law, and A. Tarakanov, SIAM/ASA J. Uncertain. Quantif., 11 (2023), pp. 616--645], which requires access only to time-discretized versions of the dynamics, i.e., the ones 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.