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Eigenfunction martingale estimators for interacting particle systems and their mean field limit

Grigorios A. Pavliotis, Andrea Zanoni

TL;DR

This paper addresses parameter inference for large systems of exchangeable interacting diffusions by observing a single particle in discrete time. It develops a novel estimator based on martingale estimating functions constructed from the eigenvalues and eigenfunctions of the mean-field generator, using the invariant measure at equilibrium to circumvent the time-varying law. The authors prove that the estimator is asymptotically unbiased and normal as the number of observations and particles grows, and they provide a concrete convergence rate and an explicit asymptotic covariance, supported by extensive numerical experiments. The results demonstrate robustness to sampling rate, competitive performance against MLE, and applicability even in regimes with multiple stationary states, highlighting practical utility for learning mean-field drift parameters in complex systems.

Abstract

We study the problem of parameter estimation for large exchangeable interacting particle systems when a sample of discrete observations from a single particle is known. We propose a novel method based on martingale estimating functions constructed by employing the eigenvalues and eigenfunctions of the generator of the mean field limit, where the law of the process is replaced by the (unique) invariant measure of the mean field dynamics. We then prove that our estimator is asymptotically unbiased and asymptotically normal when the number of observations and the number of particles tend to infinity, and we provide a rate of convergence towards the exact value of the parameters. Finally, we present several numerical experiments which show the accuracy of our estimator and corroborate our theoretical findings, even in the case the mean field dynamics exhibit more than one steady states.

Eigenfunction martingale estimators for interacting particle systems and their mean field limit

TL;DR

This paper addresses parameter inference for large systems of exchangeable interacting diffusions by observing a single particle in discrete time. It develops a novel estimator based on martingale estimating functions constructed from the eigenvalues and eigenfunctions of the mean-field generator, using the invariant measure at equilibrium to circumvent the time-varying law. The authors prove that the estimator is asymptotically unbiased and normal as the number of observations and particles grows, and they provide a concrete convergence rate and an explicit asymptotic covariance, supported by extensive numerical experiments. The results demonstrate robustness to sampling rate, competitive performance against MLE, and applicability even in regimes with multiple stationary states, highlighting practical utility for learning mean-field drift parameters in complex systems.

Abstract

We study the problem of parameter estimation for large exchangeable interacting particle systems when a sample of discrete observations from a single particle is known. We propose a novel method based on martingale estimating functions constructed by employing the eigenvalues and eigenfunctions of the generator of the mean field limit, where the law of the process is replaced by the (unique) invariant measure of the mean field dynamics. We then prove that our estimator is asymptotically unbiased and asymptotically normal when the number of observations and the number of particles tend to infinity, and we provide a rate of convergence towards the exact value of the parameters. Finally, we present several numerical experiments which show the accuracy of our estimator and corroborate our theoretical findings, even in the case the mean field dynamics exhibit more than one steady states.
Paper Structure (18 sections, 10 theorems, 119 equations, 10 figures, 1 algorithm)

This paper contains 18 sections, 10 theorems, 119 equations, 10 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main contributions.
  3. Outline.
  4. Problem setting
  5. Parameter estimation problem
  6. Main results
  7. Numerical experiments
  8. Sensitivity analysis and rate of convergence
  9. Comparison with the maximum likelihood estimator
  10. Diffusion coefficient
  11. Central limit theorem
  12. Double well potential
  13. Nonsymmetric confining potential
  14. Proof of the main results
  15. Limits of the estimating function and its derivative
  16. ...and 3 more sections

Key Result

Theorem 2.9

Let $J$ be a positive integer and let $\{ \widetilde{X}_m^{(n)} \}_{m = 1}^M$ be a set of observations obtained by system eq:SDE_N with true parameter $\theta_0$. Under ass:potentialass:functions_psi and if there exists $N_0 > 0$ such that for all $N > N_0$ an estimator $\widehat{\theta}_{M,N}^J$, which solves the system $G_{M,N}^J(\theta) = 0$, exists with probability tending to one as $M$ goes

Figures (10)

  • Figure 1: Sensitivity analysis for the Ornstein--Uhlenbeck potential with respect to the number $M$ of observations and $N$ of particles, for the estimator $\widehat{\theta}^J_{M,N}$ with $J=1$.
  • Figure 2: Sensitivity analysis for the Ornstein--Uhlenbeck potential with respect to the number $J$ of eigenvalues and eigenfunctions, for the estimator $\widehat{\theta}^J_{M,N}$.
  • Figure 3: Rates of convergence for the Ornstein--Uhlenbeck potential with respect to the number $M$ of observations and $N$ of particles, for the estimator $\widehat{\theta}^J_{M,N}$ with $J=1$.
  • Figure 4: Comparison between the estimator $\widehat{\theta}^J_{M,N}$ with $J=1$ (left) and the maximum likelihood estimator $\widetilde{\theta}_{M,N}^{\mathrm{MLE}}$ (right) varying the distance $\Delta$ between two consecutive observations for the Ornstein--Uhlenbeck potential.
  • Figure 5: Inference of the diffusion coefficient based on the quadratic variation varying the distance $\Delta$ between two consecutive observations for the Ornstein--Uhlenbeck potential.
  • ...and 5 more figures

Theorems & Definitions (30)

  • Remark 2.1
  • Example 2.3
  • Remark 2.4
  • Remark 2.6
  • Remark 2.7
  • Example 2.8
  • Theorem 2.9
  • Theorem 2.10
  • Theorem 2.11
  • Remark 2.12
  • ...and 20 more