A proximal gradient algorithm for composite log-concave sampling

2026-05-12

AI Summary

This work addresses sampling from composite log-concave distributions of the form π ∝ e^{−f−g}, where f is smooth and g may be non-smooth but admits a restricted Gaussian oracle (RGO). It introduces, for the first time, the proximal gradient paradigm from optimization into sampling, proposing an efficient MCMC algorithm that combines gradients of f with the RGO for g. Under strong convexity of f+g and smoothness of f, the algorithm achieves total variation error ε in Õ(κ√d log⁴(1/ε)) iterations, matching the optimal complexity when g=0. The approach is further extended to non-log-concave settings satisfying Poincaré or log-Sobolev inequalities, as well as to cases where f itself may be non-smooth.

Abstract

We propose an algorithm to sample from composite log-concave distributions over $\mathbb{R}^d$, i.e., densities of the form $π\propto e^{-f-g}$, assuming access to gradient evaluations of $f$ and a restricted Gaussian oracle (RGO) for $g$. The latter requirement means that we can easily sample from the density $\text{RGO}_{g,h,y}(x) \propto \exp(-g(x) -\frac{1}{2h}||y-x||^2)$, which is the sampling analogue of the proximal operator for $g$. If $f + g$ is $α$-strongly convex and $f$ is $β$-smooth, our sampler achieves $\varepsilon$ error in total variation distance in $\widetilde{\mathcal O}(κ\sqrt d \log^4(1/\varepsilon))$ iterations where $κ:= β/α$, which matches prior state-of-the-art results for the case $g=0$. We further extend our results to cases where (1) $π$ is non-log-concave but satisfies a Poincaré or log-Sobolev inequality, and (2) $f$ is non-smooth but Lipschitz.

Problem

• composite log-concave sampling
• proximal gradient
• restricted Gaussian oracle
• non-log-concave sampling
• total variation distance

Innovation

• proximal gradient sampling
• composite log-concave distribution
• restricted Gaussian oracle
• strong convexity
• Poincaré inequality