Rényi Tracking Bounds for Langevin Dynamics with Moving Targets

Yuchen Xin; Jingxin Zhan; Zhihua Zhang

stat.MLAug 25, 2026

Rényi Tracking Bounds for Langevin Dynamics with Moving Targets

Authors: Yuchen Xin, Jingxin Zhan, Zhihua Zhang

Organizations: School of Mathematical Sciences, Peking University

Abstract

We study Langevin diffusion and Langevin Monte Carlo (LMC) when the target distribution changes over time. Under a log-Sobolev inequality (LSI), we derive non-asymptotic Rényi-divergence guarantees for tracking the current target. The framework covers continuous-time Langevin diffusion and its discretizations. We then apply the results to nonsmooth sampling based on successive Moreau envelopes. For this scheme, we give explicit choices of the smoothing parameters and step sizes, together with corresponding complexity bounds. To our knowledge, these are the first non-asymptotic Rényi-divergence tracking bounds for Langevin dynamics with discrete target updates.

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Reflected Anchored Langevin Algorithms

Changwei Tu, Xiaoyu Wang, Yingli Wang, Xicheng Zhang+1

Hong Kong University of Science and Technology (Guangzhou), Guangzhou, Guangdong Province, People’s Republic of China · School of Mathematical Sciences, Fudan University, Shanghai, People’s Republic of China · School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, People’s Republic of China+1

Langevin Dynamics Markov Chain Monte Carlo

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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.

Abstract

Explore similar work

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