Efficient Online Learning in Interacting Particle Systems

TL;DR

This paper presents a new online parameter estimation method for stochastic interacting particle systems, using continuous observations and stochastic approximation, with proven convergence and practical numerical examples.

Contribution

The paper introduces a recursive online estimation technique for interacting particle systems, with rigorous convergence analysis and applicability to various models.

Findings

Method converges to stationary points of the asymptotic log-likelihood.

Establishes L^2 convergence rate and central limit theorem.

Numerical examples demonstrate effectiveness in practical models.

Abstract

We introduce a new method for online parameter estimation in stochastic interacting particle systems, based on continuous observation of a small number of particles from the system. Our method recursively updates the model parameters using a stochastic approximation of the gradient of the asymptotic log likelihood, which is computed using the continuous stream of observations. Under suitable assumptions, we rigorously establish convergence of our method to the stationary points of the asymptotic log-likelihood of the interacting particle system. We consider asymptotics both in the limit as the time horizon $t\rightarrow\infty$, for a fixed and finite number of particles, and in the joint limit as the number of particles $N\rightarrow\infty$ and the time horizon $t\rightarrow\infty$. Under additional assumptions on the asymptotic log-likelihood, we also establish an $\mathrm{L}^2$ convergence rate and a central limit theorem. Finally, we present several numerical examples of practical interest, including a model for systemic risk, a model of interacting FitzHugh--Nagumo neurons, and a Cucker--Smale flocking model. Our numerical results corroborate our theoretical results, and also suggest that our estimator is effective even in cases where the assumptions required for our theoretical analysis do not hold.