A Proximal Algorithm for Sampling

We consider sampling problems with possibly non-smooth potentials (negative log-densities). In particular, we study two specific settings of sampling where the convex potential is either semi-smooth or in composite form as the sum of a smooth component and a semi-smooth component. To overcome the challenges caused by the non-smoothness, we propose a Markov chain Monte Carlo algorithm that resembles proximal methods in optimization for these sampling tasks. The key component of our method is a sampling scheme for a quadratically regularized target potential. This scheme relies on rejection sampling with a carefully designed Gaussian proposal whose center is an approximate minimizer of the regularized potential. We develop a novel technique (a modified Gaussian integral) to bound the complexity of this rejection sampling scheme in spite of the non-smoothness in the potentials. We then combine this scheme with the alternating sampling framework (ASF), which uses Gibbs sampling on an augmented distribution, to accomplish the two settings of sampling tasks we consider. Furthermore, by combining the complexity bound of the rejection sampling we develop and the remarkable convergence properties of ASF discovered recently, we are able to establish several non-asymptotic complexity bounds for our algorithm, in terms of the total number of queries of subgradient of the target potential. Our algorithm achieves state-of-the-art complexity bounds compared with all existing methods in the same settings.