A Proximal Algorithm for Sampling from Non-Smooth Potentials

In this work, we examine sampling problems with non-smooth potentials and propose a novel Markov chain Monte Carlo algorithm for it. We provide a non-asymptotical analysis of our algorithm and establish a polynomial-time complexity
$\tilde{\mathscr{O}}(M^{2}d_{4}\mathscr{M}^{1/2}_{4}\varepsilon^{-1})$
to achieve
$\varepsilon$
error in terms of total variation distance to a log-concave target density with 4th moment
$\mathscr{M}_{4}$
and M-Lipschitz potential, better than most existing results under the same assumptions. Our method is based on the proximal bundle method and an alternating sampling framework. The latter framework requires the so-called restricted Gaussian oracle, which can be viewed as a sampling counterpart of the proximal mapping in convex optimization. One key contribution of this work is a fast algorithm that realizes the restricted Gaussian oracle for any convex non-smooth potential with bounded Lipschitz constant.