A proximal gradient algorithm for composite log-concave sampling

We propose an algorithm to sample from composite log-concave distributions over ℝd, i.e., densities of the form πpropto 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 RGOg,h,y(x) propto exp(-g(x) -frac12h||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 widetildemathcal O(kappa √ d log4(1/varepsilon)) iterations where kappa := 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 Poincar'e or log-Sobolev inequality, and (2) f is non-smooth but Lipschitz.