Zeroth-Order Non-Log-Concave Sampling with Variance Reduction and Applications to Inverse Problems

2026-05-28

AI Summary

Efficient sampling from high-dimensional non-log-concave distributions remains challenging in black-box settings, as existing zeroth-order methods suffer from high variance and lack non-asymptotic convergence guarantees. This work proposes a variance-reduced zeroth-order Langevin sampling algorithm (ZO-APMC) that employs an improved gradient estimator to reduce variance and eliminate the dependence of batch size on dimensionality. For the first time, non-asymptotic convergence guarantees for zeroth-order sampling from non-log-concave distributions are established, measured in terms of ε-relative Fisher information and total variation distance. Building upon this theoretical foundation, the authors develop the first black-box posterior sampling algorithm for inverse problems with such guarantees. Experiments demonstrate that the proposed method achieves superior sampling performance and stability on both synthetic data and real-world linear and nonlinear inverse problems.

Abstract

Sampling from high-dimensional, non-log-concave distributions with unnormalized densities remains a fundamental challenge in machine learning, particularly in black-box settings where gradient information is inaccessible or computationally prohibitive. While Langevin dynamics provides a principled framework for sampling when gradients are accessible, its extension to the black-box settings suffers from high variance and lacks non-asymptotic convergence guarantees for non-log-concave sampling. To address these limitations, we propose a variance-reduced zeroth-order Langevin sampling method. Our method employs a gradient estimator that substantially reduces the variance of the classical batched zeroth-order estimator and eliminates the unfavorable dimensional dependence of the batch size required for accurate estimation, enabling practical and stable sampling. We establish the first non-asymptotic convergence guarantees for zeroth-order non-log-concave sampling in terms of $\varepsilon$-relative Fisher information, and, under a Poincaré inequality assumption, squared total variation distance. We further propose ZO-APMC, a posterior sampling algorithm for black-box inverse problems with pre-trained score-based generative priors, establishing the first non-asymptotic convergence guarantees for such methods. We validate our theory through synthetic experiments and demonstrate strong empirical performance on practical linear and nonlinear inverse problems.

Problem

• zeroth-order sampling
• non-log-concave distributions
• black-box sampling
• inverse problems
• gradient-free sampling

Innovation

• zeroth-order sampling
• variance reduction
• non-log-concave sampling
• Langevin dynamics
• inverse problems