Variance Reduction for Non-Log-Concave Sampling with Applications to Inverse Problems

2026-06-15

AI Summary

This work addresses the inefficiency and poor sample quality in sampling from high-dimensional non-log-concave distributions, which stem from high-variance stochastic gradients. We propose the first unified sampling framework that integrates variance reduction techniques—including momentum SGD, STORM, and PAGE—into a coherent algorithmic structure. Under a fixed gradient budget, our method substantially improves both convergence speed and sample fidelity. Theoretically, we establish the first non-asymptotic analysis for variance-reduced sampling in non-log-concave settings, proving accelerated convergence rates measured in relative Fisher information and total variation distance. Empirically, we validate the effectiveness of the proposed approach on two classes of imaging inverse problems and further demonstrate its applicability to solvers leveraging score-based generative priors.

Abstract

Sampling from high-dimensional, non-log-concave distributions with unnormalized densities is a fundamental challenge in machine learning, particularly when the exact gradient of the potential is unavailable and must be approximated via stochastic gradients that exhibit high variance under a fixed budget of gradient computations per iteration. Although variance reduction techniques such as SGD with momentum, STORM, and PAGE have demonstrated improved convergence properties in non-convex optimization, their implications for sampling from non-log-concave distributions remain largely unexplored. In this work, we develop the first unified analysis of these estimators for sampling from non-log-concave distributions. We establish improved non-asymptotic convergence rates in $\varepsilon$-relative Fisher information and, under a Poincaré inequality assumption, in squared total variation distance, and further prove weak convergence to the target distribution. We extend our analysis to solving inverse problems with score-based generative priors. We empirically validate our theory and demonstrate that, under a fixed gradient computations per iteration, variance-reduction techniques consistently improve sample quality in two standard imaging applications.

Problem

• non-log-concave sampling
• variance reduction
• stochastic gradients
• inverse problems
• high-dimensional sampling

Innovation

• variance reduction
• non-log-concave sampling
• stochastic gradient
• score-based generative models
• inverse problems