Stochastic zeroth-order discretizations of Langevin diffusions for Bayesian inference

Abstract

Discretizations of Langevin diffusions provide a powerful method for sampling and Bayesian inference. However, such discretizations require evaluation of the gradient of the potential function. In several real-world scenarios, obtaining gradient evaluations might either be computationally expensive, or simply impossible. In this work, we propose and analyze stochastic zeroth-order sampling algorithms for discretizing overdamped and underdamped Langevin diffusions. Our approach is based on estimating the gradients, based on Gaussian Stein’s identities, widely used in the stochastic optimization literature. We provide a comprehensive oracle complexity analysis – number noisy function evaluations to be made to obtain an ϵ-approximate sample in Wasserstein distance – of stochastic zeroth-order discretizations of both overdamped and underdamped Langevin diffusions, under various noise models. Our theoretical contributions extend the applicability of sampling algorithms to the noisy black-box settings arising in practice.