Nonparametric estimation for interacting particle systems : McKean-Vlasov models

TL;DR

This paper develops nonparametric kernel estimators for McKean-Vlasov models based on observed particle trajectories, providing theoretical guarantees and optimality results for large populations.

Contribution

It introduces new adaptive estimators for the nonlinear Fokker-Planck solution and interaction drift, with a novel concentration inequality for McKean-Vlasov empirical measures.

Findings

Established oracle inequalities for the estimators.

Proved minimax optimality over anisotropic H"older classes.

Derived consistent estimators for the interaction potential in Vlasov models.

Abstract

We consider a system of $N$ interacting particles, governed by transport and diffusion, that converges in a mean-field limit to the solution of a McKean-Vlasov equation. From the observation of a trajectory of the system over a fixed time horizon, we investigate nonparametric estimation of the solution of the associated nonlinear Fokker-Planck equation, together with the drift term that controls the interactions, in a large population limit $N \rightarrow \infty$. We build data-driven kernel estimators and establish oracle inequalities, following Lepski's principle. Our results are based on a new Bernstein concentration inequality in McKean-Vlasov models for the empirical measure around its mean, possibly of independent interest. We obtain adaptive estimators over anisotropic H\"older smoothness classes built upon the solution map of the Fokker-Planck equation, and prove their optimality in a minimax sense. In the specific case of the Vlasov model, we derive an estimator of the interaction potential and establish its consistency.