Stochastic Zeroth-order Discretizations of Langevin Diffusions for Bayesian Inference (1902.01373v4)

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 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.

• The paper presents stochastic approximation techniques that bypass direct gradient evaluations using Gaussian Stein's identities for black-box scenarios.
• It analyzes sample complexity in Wasserstein distance for both overdamped and underdamped Langevin dynamics under various noise models.
• The research introduces a high-dimensional variable selection mechanism that leverages gradient sparsity to optimize computational efficiency.

Stochastic Zeroth-order Discretizations of Langevin Diffusions for Bayesian Inference

The research paper explores the theoretical foundations and practical implications of utilizing stochastic zeroth-order methods for Bayesian inference via Langevin diffusions. Specifically, it examines discretizations of overdamped and underdamped Langevin dynamics without requiring direct gradient evaluations, addressing a common computational challenge in high-dimensional and noisy environments where gradient computations may be infeasible.

Zeroth-Order Sampling Algorithms

The authors propose stochastic zeroth-order methods for discretizing Langevin diffusions, which are pivotal for sampling from complex, potentially high-dimensional distributions. These methods leverage Gaussian Stein's identities, enabling them to estimate gradients indirectly through function evaluations. Such techniques sidestep the direct computation of gradients, allowing their application in black-box scenarios where analytic forms of the objective functions are unavailable or gradients are expensive to compute.

Theoretical Contributions

The paper provides an extensive analysis of the sample complexity for achieving $\epsilon$-approximate samples in Wasserstein distance. It distinguishes between overdamped and underdamped Langevin dynamics, evaluating their performance under various noise models. The results demonstrate that under appropriate settings, zeroth-order methods can match the first-order iteration complexities, albeit with a trade-off in oracle complexity, particularly in high-dimensional spaces.

High-Dimensional Variable Selection

Additionally, the paper introduces a variable selection mechanism to further enhance the efficiency of zeroth-order sampling in high dimensions. By leveraging estimated gradients' sparsity properties, the proposed method effectively reduces dimensionality during sampling, preserving computational resources while maintaining theoretical guarantees.

Implications and Future Directions

The paper's findings significantly broaden the applicability of Langevin-based sampling algorithms in settings typical to Bayesian inference and machine learning, where only noisy, zeroth-order information is available. These advances suggest promising new directions for developing more robust sampling approaches, particularly in automated and high-dimensional Bayesian modeling contexts.

The implications extend beyond immediate applications, potentially impacting areas like probabilistic modeling, uncertainty quantification, and optimization in the field of machine learning. As stochastic zeroth-order methods mature, future research may focus on refining these techniques, exploring adaptive strategies, and expanding their utility across other function classes and noise regimes.

This body of work represents a substantial step toward more flexible and computationally efficient approaches for probabilistic inference, making a solid theoretical contribution to the field of Bayesian statistics and stochastic processes. Future studies are encouraged to build on these insights, possibly integrating these methods with other machine learning paradigms or large-scale data scenarios to push the boundaries of current capabilities in AI-driven inference systems.