A proximal gradient algorithm for composite log-concave sampling
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866913119134023680 |
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| author | Liu, Linghai Chewi, Sinho |
| author_facet | Liu, Linghai Chewi, Sinho |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_12461 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A proximal gradient algorithm for composite log-concave sampling Liu, Linghai Chewi, Sinho Statistics Theory Data Structures and Algorithms Machine Learning 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. |
| title | A proximal gradient algorithm for composite log-concave sampling |
| topic | Statistics Theory Data Structures and Algorithms Machine Learning |
| url | https://arxiv.org/abs/2605.12461 |