Variance Reduction for Non-Log-Concave Sampling with Applications to Inverse Problems
For practitioners of Bayesian inference and inverse problems, this work provides theoretically grounded variance reduction methods that improve sampling efficiency under limited gradient budgets.
The paper develops the first unified analysis of variance reduction techniques (SGD with momentum, STORM, PAGE) for sampling from non-log-concave distributions, establishing improved non-asymptotic convergence rates in ε-relative Fisher information and squared total variation distance, and demonstrates consistent sample quality improvements in imaging inverse problems.
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.