Title: Dynamical mean-field analysis of adaptive Langevin diffusions: Propagation-of-chaos and convergence of the linear response Show Chinese title

Title: 自适应Langevin扩散的动力学平均场分析：混沌传播和线性响应的收敛

Abstract: 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. Show Chinese abstract

Abstract: 受高维回归中经验贝叶斯学习应用的启发，我们研究了一类在随机无序系统中的Langevin扩散过程，其中漂移系数由一个参数驱动，该参数连续适应到迄今为止实现过程的经验分布。 由此产生的动力学形式为具有麦凯恩-弗拉索夫型交互作用和由随机无序定义的成对交互作用的随机相互粒子系统。 我们证明了混沌传播的结果，表明在系统维度无关的时间范围内，Langevin过程样本路径的经验分布收敛到一个确定性的极限定律，该定律由动态平均场理论描述。 此定律由极限漂移参数和极限动力学的相关性和响应核的动态固定点方程系统表征。 使用动态空腔论证，我们验证了这些相关性和响应核作为系统的单个坐标平均相关性和线性响应函数的渐近极限出现。 这些结果使我们能够对线性回归模型中未知先验参数的经验贝叶斯Langevin动力学程序进行渐近分析，我们在另一篇配套论文中开发了这一程序。