发表机构
Hong Kong University of Science and Technology (Guangzhou); Fudan University; Beijing Institute of Technology; Florida State University（香港科技大学（广州）; 复旦大学; 北京理工大学; 佛罗里达州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出反射锚定朗之万算法，通过平滑锚定参考势处理不可微目标，实现约束域上的高效采样，并给出收敛保证与实验验证。
AI 中文摘要
机器学习中用于约束采样的二阶朗之万算法，例如基于反射朗之万动力学离散化的投影朗之万蒙特卡洛，需要可微的对数密度，这限制了它们的适用性。本文介绍了反射锚定朗之万动力学（RALD），一种在约束域上收敛到不可微目标的反射扩散过程。该方法使用平滑的锚定参考势，并将其反射朗之万动力学的漂移和噪声协方差乘以相同的状态依赖缩放因子。其带投影的欧拉-丸山离散化产生了反射锚定朗之万蒙特卡洛（RALMC）算法。我们证明了RALMC在2-Wasserstein距离下到目标分布的显式收敛界和迭代复杂度。提供了数值实验来说明理论预测和该方法的实证性能。
英文摘要
First order Langevin algorithms for constrained sampling in machine learning, such as projected Langevin Monte Carlo which are based on discretizations of reflected Langevin dynamics, require differentiable log densities that limits their applicability. This paper introduces reflected anchored Langevin dynamics (RALD), a reflected diffusion that converges to non-differentiable targets on constrained domains. The method uses a smooth anchored reference potential and multiplies the drift and noise covariance of its reflected Langevin dynamics by the same state dependent scaling factor. Its Euler-Maruyama discretization with projection gives reflected anchored Langevin Monte Carlo (RALMC) algorithm. We prove explicit convergence bounds and iteration complexity for RALMC in the 2-Wasserstein distance to the target distribution. Numerical experiments are provided to illustrate the theoretical predictions and the empirical performance of the method.
Comments70 pages, 9 figures
