Reflected Anchored Langevin Algorithms
Abstract
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.
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BibTeX
@misc{tu2026reflected,
title = {{Reflected Anchored Langevin Algorithms}},
author = {Changwei Tu and Xiaoyu Wang and Yingli Wang and Xicheng Zhang and Lingjiong Zhu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09522},
eprint = {2610.09522},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09522},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Reflected Anchored Langevin Algorithms
AU - Changwei Tu
AU - Xiaoyu Wang
AU - Yingli Wang
AU - Xicheng Zhang
AU - Lingjiong Zhu
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09522
UR - https://arxiv.org/abs/2610.09522
AB - 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.
ER -