A proximal gradient algorithm for composite log-concave sampling

• URL: http://arxiv.org/abs/2605.12461v1
• Date: Tue, 12 May 2026 17:48:09 GMT
• Title: A proximal gradient algorithm for composite log-concave sampling
• Abstract summary: We propose an algorithm to sample from composite log-concave distributions over $mathbbRd$, i.e., densities of the form $propto e-f-g$.<n>We further extend our results to cases where (1) $$ is non-log-concave but satisfies a Poincaré or log-Sobolev inequality, and (2) $f$ is non-smooth but Lipschitz.
• Score: 9.11122093402205
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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