通过大偏差理论优化约束采样，加速收敛并降低方差。
Accelerating Constrained Sampling: A Large Deviations Approach
- 引入斜对称矩阵设计，使边界法向量乘积为零，提升采样效率。
- 理论证明该设计可加速收敛，降低渐近方差，优于传统反射动力学。
- 数值实验验证方法在约束采样任务中表现更优，适合高维概率采样场景。
约束域上目标概率分布的采样问题广泛存在于机器学习等领域。已有研究提出基于反射Langevin动力学（RLD）的投影Langevin蒙特卡洛（PLMC）及更一般的斜反射非可逆Langevin蒙特卡洛（SRNLMC），其基于斜反射非可逆Langevin动力学（SRNLD）的离散化。本文聚焦于SRNLD的长时间行为，其中引入一个斜对称矩阵。尽管已知此类动力学可加速，但如何选择该矩阵以实现实际性能仍不明确。本文建立了当斜对称矩阵与边界外法向量场的乘积为零时，SRNLD经验测度的大偏差原理（LDP）。通过显式刻画速率函数，证明此选择能加速收敛至目标分布，并减少渐近方差。基于该设计的SRNLMC数值实验显示优异性能，验证了大偏差理论的结论。
原文摘要 · Abstract (English)
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.
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