Adaptive Langevin diffusion的动力学均值场分析：Replica对称固定点与经验贝叶斯
摘要
在通过采样进行统计估计的许多应用中，可能希望从适应于已见样本而演化的高维目标分布中进行采样。我们研究了这样一种动力学的例子，即针对具有独立同分布回归设计的贝叶斯线性回归模型后验采样的Langevin扩散，其先验通过最大边际似然方案持续适应Langevin轨迹。我们在伴随论文中发展的动力学均值场理论(DMFT)结果，establish a precise high-dimensional asymptotic limit for the joint evolution of the prior parameter and law of the Langevin sample. In this work, we carry out an analysis of the equations that describe this DMFT limit, under conditions of approximate time-translation-invariance which include, in particular, settings where the posterior law satisfies a log-Sobolev inequality. In such settings, we show that this adaptive Langevin trajectory converges on a dimension-independent time horizon to an equilibrium state that is characterized by a system of scalar fixed-point equations, and the associated prior parameter converges to a critical point of a replica-symmetric limit for the model free energy. As a by-product of our analyses, we obtain a new dynamical proof that this replica-symmetric limit for the free energy is exact, in models having a possibly misspecified prior and where a log-Sobolev inequality holds for the posterior law.
引用
@article{arxiv.2504.15558,
title = {Dynamical mean-field analysis of adaptive Langevin diffusions: Replica-symmetric fixed point and empirical Bayes},
author = {Zhou Fan and Justin Ko and Bruno Loureiro and Yue M. Lu and Yandi Shen},
journal= {arXiv preprint arXiv:2504.15558},
year = {2025}
}