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.

翻译：我们提出一种算法，用于从 $\mathbb{R}^d$ 上的复合对数凹分布中采样，即密度形式为 $π\propto e^{-f-g}$，假设可获取 $f$ 的梯度评估以及 $g$ 的受限高斯预言机（RGO）。后者要求我们能轻松从密度 $\text{RGO}_{g,h,y}(x) \propto \exp(-g(x) -\frac{1}{2h}||y-x||^2)$ 中采样，该密度对应于 $g$ 的邻近算子的采样类比。若 $f + g$ 为 $α$-强凸且 $f$ 为 $β$-光滑，本采样器在 $\widetilde{\mathcal O}(κ\sqrt d \log^4(1/\varepsilon))$ 次迭代内实现总变差距离上的 $\varepsilon$ 误差，其中 $κ:= β/α$，与 $g=0$ 情形下先前最先进结果相匹配。我们进一步将结果推广至以下情形：(1) $π$ 非对数凹但满足庞加莱或对数索伯列夫不等式；(2) $f$ 非光滑但满足利普希茨条件。