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.

Questo articolo esplora i giri e le loro implicazioni.

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