In this paper, we study the problem of sampling from log-concave distributions supported on convex and compact sets, with a particular focus on the randomized midpoint discretization of both overdamped and kinetic Langevin diffusions in constrained domains. We revisit the proximal framework for handling constraints through projection operators and develop a more general formulation that encompasses Euclidean, Bregman, and Gauge projections. The resulting smooth approximation allows a unified and tractable analysis of Langevin algorithms and their variants under constraints. Within this framework, we establish convergence guarantees in Wasserstein-$q$ $(q\geqslant 1)$ distances between the smooth surrogate and the target distribution. We further derive complementary lower bounds, showing that the results are near-optimal in order. Building upon this tight approximation analysis, we obtain new convergence guarantees for the randomized midpoint Langevin algorithms and refined bounds for both vanilla and kinetic Langevin Monte Carlo methods under constraints, thereby advancing the theoretical understanding of constrained diffusion-based sampling.

翻译：本文研究在凸紧集支撑下的对数凹分布采样问题，重点探讨约束域中超阻尼与动Langevin扩散的随机中点离散化方法。我们重新审视了通过投影算子处理约束的近端框架，并发展出涵盖欧几里得投影、Bregman投影和Gauge投影的更一般性框架。所得到的平滑近似方法能够对约束条件下的Langevin算法及其变体进行统一且易处理的解析。在此框架下，我们建立了平滑代理分布与目标分布之间Wasserstein-$q$ $(q\geqslant 1)$ 距离的收敛性保证，并进一步推导出互为补充的下界，表明该结果在阶数上接近最优。基于这种紧致近似分析，我们获得了约束条件下随机中点Langevin算法的全新收敛性保证，以及对原始Langevin蒙特卡洛方法和动Langevin蒙特卡洛方法的改进界，从而推动了约束扩散采样理论的发展。