On nonparametric estimation of the interaction function in particle system models

TL;DR

This paper proposes two nonparametric estimation methods for the interaction function in particle system models, analyzing their errors and showing minimax rates of convergence under certain metrics, which is surprising given the problem's complexity.

Contribution

Introduces two empirical risk minimization-based estimation methods and establishes minimax convergence rates for the interaction function in particle systems.

Findings

Minimax estimation error converges at a parametric rate.

Both methods effectively control stochastic and approximation errors.

Surprising result: parametric rate holds for large classes of interaction functions.

Abstract

This paper delves into a nonparametric estimation approach for the interaction function within diffusion-type particle system models. We introduce two estimation methods based upon an empirical risk minimization. Our study encompasses an analysis of the stochastic and approximation errors associated with both procedures, along with an examination of certain minimax lower bounds. In particular, we show that there is a natural metric under which the corresponding minimax estimation error of the interaction function converges to zero with parametric rate. This result is rather suprising given complexity of the underlying estimation problem and rather large classes of interaction functions for which the above parametric rate holds.