Randomized Midpoint Method for Log-Concave Sampling under Constraints

TL;DR

This paper introduces a randomized midpoint method for sampling from log-concave distributions on convex sets, providing new convergence guarantees and complexity bounds for constrained Langevin diffusions.

Contribution

It presents a unified proximal framework for constrained Langevin diffusions with non-asymptotic bounds and improved convergence guarantees.

Findings

Established non-asymptotic $ ext{W}_1$ and $ ext{W}_2$ bounds

Provided sharper convergence guarantees for Langevin Monte Carlo

Compared performance of different projection operators

Abstract

In this paper, we study the problem of sampling from log-concave distributions supported on convex, compact sets, with a particular focus on the randomized midpoint discretization of both vanilla and kinetic Langevin diffusions in this constrained setting. We propose a unified proximal framework for handling constraints via a broad class of projection operators, including Euclidean, Bregman, and Gauge projections. Within this framework, we establish non-asymptotic bounds in both $\mathcal{W}_1$ and $\mathcal{W}_2$ distances, providing precise complexity guarantees and performance comparisons. In addition, our analysis leads to sharper convergence guarantees for both vanilla and kinetic Langevin Monte Carlo under constraints, improving upon existing theoretical results.