Long run convergence of discrete-time interacting particle systems of the McKean-Vlasov type - Equipe Signal, Statistique et Apprentissage Access content directly
Preprints, Working Papers, ... Year : 2024

Long run convergence of discrete-time interacting particle systems of the McKean-Vlasov type

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.
Fichier principal
Vignette du fichier
v17.pdf (571.55 Ko) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

hal-04512759 , version 1 (20-03-2024)
hal-04512759 , version 2 (25-03-2024)
hal-04512759 , version 3 (02-04-2024)

Licence

Attribution

Identifiers

Cite

Pascal Bianchi, Walid Hachem, Victor Priser. Long run convergence of discrete-time interacting particle systems of the McKean-Vlasov type. 2024. ⟨hal-04512759v3⟩
87 View
32 Download

Altmetric

Share

Gmail Facebook X LinkedIn More