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Non-parametric estimates for graphon mean-field particle systems

Erhan Bayraktar, Hongyi Zhou

Abstract

We consider the graphon mean-field system introduced in the work of Bayraktar, Chakraborty, and Wu. It is the large-population limit of a heterogeneously interacting diffusive particle system, where the interaction is of mean-field type with weights characterized by an underlying graphon function. Through observation of continuous-time trajectories within the particle system, we construct plug-in estimators of the particle density, the drift coefficient, and thus the graphon interaction weights of the mean-field system. Our estimators for the density and drift are direct results of kernel interpolation on the empirical data, and a deconvolution method leads to an estimator of the underlying graphon function. We show that, as the number of particles increases, the graphon estimator converges to the true graphon function pointwisely, and as a consequence, in the cut metric. Besides, we conduct a minimax analysis within a particular class of particle systems to justify the pointwise optimality of the density and drift estimators.

Non-parametric estimates for graphon mean-field particle systems

Abstract

We consider the graphon mean-field system introduced in the work of Bayraktar, Chakraborty, and Wu. It is the large-population limit of a heterogeneously interacting diffusive particle system, where the interaction is of mean-field type with weights characterized by an underlying graphon function. Through observation of continuous-time trajectories within the particle system, we construct plug-in estimators of the particle density, the drift coefficient, and thus the graphon interaction weights of the mean-field system. Our estimators for the density and drift are direct results of kernel interpolation on the empirical data, and a deconvolution method leads to an estimator of the underlying graphon function. We show that, as the number of particles increases, the graphon estimator converges to the true graphon function pointwisely, and as a consequence, in the cut metric. Besides, we conduct a minimax analysis within a particular class of particle systems to justify the pointwise optimality of the density and drift estimators.
Paper Structure (22 sections, 15 theorems, 278 equations)

This paper contains 22 sections, 15 theorems, 278 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Our contributions and organization of this paper
  4. Model and estimators
  5. Setting, notation and assumptions
  6. Plug-in estimators
  7. Convergence of estimators
  8. Error bounds for density and drift
  9. Proof of main theorem
  10. Minimax analysis on plug-in estimators
  11. Anisotropic Hölder smoothness classes
  12. Minimax estimation for density
  13. Minimax estimation for drift
  14. Proofs for Section \ref{['s:estimates']}
  15. Proof of Lemma \ref{['l:JK-est-mu']} and Corollary \ref{['c:err-bound-mu-overall']}
  16. ...and 7 more sections

Key Result

proposition 1

Assume Conditions cd:coeffs(1)(3)(4), cd:init-data(1), and that $b$ is almost everywhere bounded. There exists some $C, R > 0$ such that, for every $p > d+2$ and every bounded open set $U$ disjoint from the closed ball $\overline{B(0,R)}$, we have for all $t \in (0,T)$ and $u \in I$ that As a consequence,

Theorems & Definitions (40)

  • proposition 1
  • lemma 1: Theorem 3.2, BayraktarChakrabortyWu2023
  • theorem 1: Main theorem
  • remark 1
  • remark 2
  • lemma 2
  • corollary 1
  • lemma 3
  • corollary 2
  • proof : Proof of Theorem \ref{['t:main']}
  • ...and 30 more