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 $α_f$-convex $f$ and a restricted Gaussian oracle (RGO) for $α_g$-convex $g$, where $α_g \ge 0$. 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 $α$-convex and $f$ is $β$-smooth, our sampler achieves $\varepsilon$ error in total variation distance using $\widetilde{\mathcal O}(κ\sqrt d \log^4(1/\varepsilon))$ oracle calls where $κ:= 1 \vee β/α$, with $0\le α_g \le β$ and $α:= α_f + α_g > 0$, 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：暂无翻译