arXiv：cs.LG· Linghai Liu, Sinho Chewi·· 6 小时前
面向复合对数凹采样的近端梯度算法
A proximal gradient algorithm for composite log-concave sampling
AI 导读
研究者提出一种从复合对数凹分布 π∝e^{-f-g} 中采样的近端梯度算法，假设可访问 α_f-凸函数 f 的梯度，并拥有 α_g-凸函数 g 的受限高斯 oracle（RGO）。
正文
Abstract:We propose an algorithm to sample from composite log-concave distributions over $\mathbb{R}^d$, i.e., densities of the form $\pi\propto e^{-f-g}$, assuming access to gradient evaluations of $\alpha_f$-convex $f$ and a restricted Gaussian oracle (RGO) for $\alpha_g$-convex $g$, where $\alpha_g \ge 0$. 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 $\alpha$-convex and $f$ is $\beta$-smooth, our sampler achieves $\varepsilon$ error in total variation distance using $\widetilde{\mathcal O}(\kappa \sqrt d \log^4(1/\varepsilon))$ oracle calls where $\kappa := 1 \vee \beta/\alpha$, with $0\le \alpha_g \le \beta$ and $\alpha := \alpha_f + \alpha_g > 0$, which matches prior state-of-the-art results for the case $g=0$. We further extend our results to cases where (1) $\pi$ is non-log-concave but satisfies a Poincaré or log-Sobolev inequality, and (2) $f$ is non-smooth but Lipschitz.
| Subjects: | Statistics Theory (math.ST); Data Structures and Algorithms (cs.DS); Machine Learning (cs.LG); Machine Learning (stat.ML) |
| Cite as: | arXiv:2605.12461 [math.ST] |
| (or arXiv:2605.12461v2 [math.ST] for this version) | |
| https://doi.org/10.48550/arXiv.2605.12461 arXiv-issued DOI via DataCite |
Submission history
From: Linghai Liu [view email]
[v1]
Tue, 12 May 2026 17:48:09 UTC (121 KB)
[v2]
Wed, 14 Oct 2026 09:23:20 UTC (124 KB)
来源：arXiv：cs.LG · arxiv.org