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.