Motivated by an application to empirical Bayes learning in high-dimensional regression, we study a class of Langevin diffusions in a system with random disorder, where the drift coefficient is driven by a parameter that continuously adapts to the empirical distribution of the realized process up to the current time. The resulting dynamics take the form of a stochastic interacting particle system having both a McKean-Vlasov type interaction and a pairwise interaction defined by the random disorder. We prove a propagation-of-chaos result, showing that in the large system limit over dimension-independent time horizons, the empirical distribution of sample paths of the Langevin process converges to a deterministic limit law that is described by dynamical mean-field theory. This law is characterized by a system of dynamical fixed-point equations for the limit of the drift parameter and for the correlation and response kernels of the limiting dynamics. Using a dynamical cavity argument, we verify that these correlation and response kernels arise as the asymptotic limits of the averaged correlation and linear response functions of single coordinates of the system. These results enable an asymptotic analysis of an empirical Bayes Langevin dynamics procedure for learning an unknown prior parameter in a linear regression model, which we develop in a companion paper.

翻译：受高维回归中经验贝叶斯学习应用的启发，我们研究了一类具有随机无序性的朗之万扩散系统，其中漂移系数由一个参数驱动，该参数持续适应于截至当前时刻已实现过程的经验分布。由此产生的动力学表现为一个具有McKean-Vlasov型相互作用和由随机无序性定义的对偶相互作用的随机相互作用粒子系统。我们证明了一个混沌传播结果：在维度无关的时间尺度上，当系统规模趋于无穷大时，朗之万过程样本路径的经验分布收敛于一个由动力学平均场理论描述的确定性极限律。该极限律通过一组动力学定点方程刻画，这些方程描述了漂移参数极限以及极限动力学的关联核与响应核。利用动力学空腔论证，我们验证了这些关联核与响应核确实对应于系统单坐标平均关联函数与线性响应函数的渐近极限。这些结果为线性回归模型中学习未知先验参数的经验贝叶斯朗之万动力学过程提供了渐近分析基础，相关方法将在我们的姊妹篇论文中展开论述。