Stochastic Zeroth-order Discretizations of Langevin Diffusions for Bayesian Inference
This work extends the practical applicability of sampling algorithms to noisy black-box and high-dimensional settings, addressing a bottleneck in real-world scenarios where gradients are inaccessible.
The paper tackles the problem of sampling and Bayesian inference when gradient evaluations are unavailable or expensive by proposing stochastic zeroth-order discretizations of Langevin diffusions, providing sample complexity analysis for obtaining ε-approximate samples in Wasserstein distance under various noise models.
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 $ε$-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.