A proximal gradient algorithm for composite log-concave sampling
TL;DR
The results are extended to cases where (1) $\pi$ is non-log-concave but satisfies a Poincar\'e or log-Sobolev inequality, and (2) $f$ is non-smooth but Lipschitz.
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 Poincar\'e or log-Sobolev inequality, and (2) $f$ is non-smooth but Lipschitz.
