Sampling from constrained statistical distributions is a fundamental task in various fields including Bayesian statistics, computational chemistry, and statistical physics. This article considers the cases where the constrained distribution is described by an unconstrained density, as well as additional equality and/or inequality constraints, which often make the constraint set nonconvex. Existing methods for nonconvex constraint set $\Sigma \subset \mathbb{R}^d$ defined by equality or inequality constraints commonly rely on costly projection steps. Moreover, they cannot handle equality and inequality constraints simultaneously as each method only specialized in one case. In addition, rigorous and quantitative convergence guarantee is often lacking. In this paper, we introduce Overdamped Langevin with LAnding (OLLA), a new framework that can design overdamped Langevin dynamics accommodating both equality and inequality constraints. The proposed dynamics also deterministically corrects trajectories along the normal direction of the constraint surface, thus obviating the need for explicit projections. We show that, under suitable regularity conditions on the target density and $\Sigma$, OLLA converges exponentially fast in $W_2$ distance to the constrained target density $\rho_\Sigma(x) \propto \exp(-f(x))d\sigma_\Sigma$. Lastly, through experiments, we demonstrate the efficiency of OLLA compared to projection-based constrained Langevin algorithms and their slack variable variants, highlighting its favorable computational cost and reasonable empirical mixing.

翻译：从约束统计分布中采样是贝叶斯统计、计算化学和统计物理等多个领域的基础任务。本文研究约束分布由无约束密度及附加等式和/或不等式约束描述的情形，此类约束常导致约束集非凸。现有处理由等式或不等式定义的非凸约束集 $\Sigma \subset \mathbb{R}^d$ 的方法通常依赖计算代价高昂的投影步骤，且无法同时处理等式与不等式约束，因为每种方法仅适用于单一情形。此外，这些方法往往缺乏严格且定量的收敛性保证。本文提出带着陆的过阻尼朗之万动力学（OLLA），该框架能设计同时适应等式与不等式约束的过阻尼朗之万动力学。所提动力学还沿约束曲面法向确定性修正轨迹，从而避免显式投影需求。我们证明，在目标密度与 $\Sigma$ 满足适当正则性条件下，OLLA 能以 $W_2$ 距离指数级快速收敛至约束目标密度 $\rho_\Sigma(x) \propto \exp(-f(x))d\sigma_\Sigma$。最后通过实验，我们展示 OLLA 相较于基于投影的约束朗之万算法及其松弛变量变体的优越效率，突显其良好的计算成本与合理的经验混合性能。