An Efficient Sampling Algorithm for Non-smooth Composite Potentials

We consider the problem of sampling from a density of the form p(x) propto exp(-f(x)- g(x)), where f: ℝd rightarrow ℝ is a smooth and strongly convex function and g: ℝd rightarrow 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.