Long run convergence of discrete-time interacting particle systems of the McKean–Vlasov type
Year: 2025
Type: article
Abstract: We consider a discrete-time system of n coupled random vectors, a.k.a. interacting particles. The dynamics involve a vanishing step size, some random centered perturbations, and a mean vector field which induces the coupling between the particles. We study the doubly asymptotic regime where both the number of iterations and the number n of particles tend to infinity, without any constraint on the relative rates of convergence of these two parameters. We establish that the empirical measure of the interpolated trajectories of the particles converges in probability, in an ergodic sense, to the set of recurrent Mc-Kean-Vlasov distributions. A first application example is the granular media equation, where the particles are shown to converge to a critical point of the Helmholtz energy. A second example is the convergence of stochastic gradient descent to the global minimizer of the risk, in a wide two-layer neural networks using random features.
Authors: Pascal Bianchi, Walid Hachem, Victor Priser
Language: English
Subfield: Finance
Domain: Social Sciences
Open Access status: hybrid
Estimated APC paid: $3,520
OpenAlex ID: W4393064314