Black-box sampling for weakly smooth Langevin Monte Carlo using p-generalized Gaussian smoothing

TL;DR

This paper introduces a novel black-box Langevin Monte Carlo method that employs p-generalized Gaussian smoothing to enable sampling from weakly smooth, non-strongly concave distributions, broadening the applicability of Langevin-based sampling techniques.

Contribution

It generalizes Gaussian smoothing and develops a black-box LMC algorithm suitable for weakly smooth, non-strongly concave densities, expanding the scope of Langevin sampling methods.

Findings

Developed a non-strongly concave, weakly smooth black-box LMC algorithm.

Theoretical analysis of the convergence properties of the proposed method.

Applicable to a broader class of distributions beyond traditional smooth, strongly log-concave cases.

Abstract

Discretization of continuous-time diffusion processes is a widely recognized method for sampling. However, the canonical Euler-Maruyama discretization of the Langevin diffusion process, also named as Langevin Monte Carlo (LMC), studied mostly in the context of smooth (gradient-Lipschitz) and strongly log-concave densities, a significant constraint for its deployment in many sciences, including computational statistics and statistical learning. In this paper, we establish several theoretical contributions to the literature on such sampling methods. Particularly, we generalize the Gaussian smoothing, approximate the gradient using p-generalized Gaussian smoothing and take advantage of it in the context of black-box sampling. We first present a non-strongly concave and weakly smooth black-box LMC algorithm, ideal for practical applicability of sampling challenges in a general setting.