The problem of sampling a target probability distribution on a constrained domain arises in many applications including machine learning. For constrained sampling, various Langevin algorithms such as projected Langevin Monte Carlo (PLMC), based on the discretization of reflected Langevin dynamics (RLD) and more generally skew-reflected non-reversible Langevin Monte Carlo (SRNLMC), based on the discretization of skew-reflected non-reversible Langevin dynamics (SRNLD), have been proposed and studied in the literature. This work focuses on the long-time behavior of SRNLD, where a skew-symmetric matrix is added to RLD. Although acceleration for SRNLD has been studied, it is not clear how one should design the skew-symmetric matrix in the dynamics to achieve good performance in practice. We establish a large deviation principle (LDP) for the empirical measure of SRNLD when the skew-symmetric matrix is chosen such that its product with the outward unit normal vector field on the boundary is zero. By explicitly characterizing the rate functions, we show that this choice of the skew-symmetric matrix accelerates the convergence to the target distribution compared to RLD and reduces the asymptotic variance. Numerical experiments for SRNLMC based on the proposed skew-symmetric matrix show superior performance, which validate the theoretical findings from the large deviations theory.

翻译：在许多应用（包括机器学习）中，都出现了在约束域上对目标概率分布进行采样的问题。针对约束采样，文献中已经提出并研究了多种朗之万算法，例如基于反射朗之万动力学（RLD）离散化的投影朗之万蒙特卡洛（PLMC），以及更一般的基于斜反射非可逆朗之万动力学（SRNLD）离散化的斜反射非可逆朗之万蒙特卡洛（SRNLMC）。本文重点研究SRNLD的长时间行为，其中在RLD的基础上添加了一个反对称矩阵。尽管已有关于SRNLD加速的研究，但在实际应用中如何设计动力学中的反对称矩阵以获得良好性能仍不明确。我们为SRNLD的经验测度建立了大偏差原理（LDP），其中反对称矩阵的选择满足其与边界上的外单位法向量场的乘积为零。通过显式刻画率函数，我们证明与RLD相比，这种反对称矩阵的选择加速了向目标分布的收敛，并降低了渐近方差。基于所提出的反对称矩阵的SRNLMC数值实验展现了优越的性能，验证了大偏差理论的理论结果。