Title: “Unbiased Estimation using the Underdamped Langevin Dynamics”

Abstract:

“In this talk, we look at sampling (unbiasedly) from probability measures that have non-negative Lebesgue density of the form $\pi(x) \propto exp(-U(x))$, with U being differentiable, via the underdamped Langevin dynamics, which has proven to be popular lately due to applications in statistics and machine learning. In particular, we compute expectations with respect to this measure. It has been proven that in continuous time the dynamics can be constructed to admit the probability of interest as a stationary measure as time goes to infinity. However, in practice, one has to stop the dynamics at some finite time which will result in a bias. Furthermore, in order to use the underdamped Langevin dynamics one needs to apply a time-discretization scheme which will also result in a bias. We develop a novel scheme based upon doubly randomized estimation, which requires access only to time-discretized versions of the dynamics and aims to remove both biases. We provide an estimator to expectations with respect to the given probability measure and prove, under standard assumptions, that our estimator is of finite variance and has a finite expected cost.”