We consider the problem of sampling from a density of the form $p(x) propto exp(-f(x)- g(x))$, where $f: mathbb{R}^d rightarrow mathbb{R}$ is a smooth and strongly convex function and $g: mathbb{R}^d rightarrow mathbb{R}$ is a convex and Lipschitz function. We propose a new algorithm based on the Metropolis-Hastings framework, and prove that it mixes to within TV distance $varepsilon$ of the target density in at most $O(d log (d/varepsilon))$ iterations. This guarantee extends previous results on sampling from distributions with smooth log densities ($g = 0$) to the more general composite non-smooth case, with the same mixing time up to a multiple of the condition number. Our method is based on a novel proximal-based proposal distribution that can be efficiently computed for a large class of non-smooth functions $g$.
Download