  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 sample complexity analysis -- number
noisy function evaluations to be made to obtain an $\epsilon$-approximate
sample in Wasserstein distance -- of stochastic zeroth-order discretizations of
both overdamped and underdamped Langevin diffusions, under various noise
models. We also propose a variable selection technique based on zeroth-order
gradient estimates and establish its theoretical guarantees. Our theoretical
contributions extend the practical applicability of sampling algorithms to the
noisy black-box and high-dimensional settings.
