Efficient Online Learning in Interacting Particle Systems

2026-02-24

AI Summary

This study addresses the problem of efficient online parameter estimation in stochastic interacting particle systems using only continuous observations of a small subset of particles. The authors propose a stochastic approximation algorithm based on the asymptotic gradient of the log-likelihood, which recursively updates model parameters. For the first time, they establish convergence, L² convergence rates, and a central limit theorem for the parameter estimates under both fixed population size and a joint asymptotic regime where the number of particles and observation time simultaneously tend to infinity. The method is validated through applications to a systemic risk model, FitzHugh–Nagumo neuronal networks, and the Cucker–Smale flocking model. Numerical experiments confirm theoretical predictions and demonstrate robustness even when key modeling assumptions are violated.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation
• Machine Learning: Online Learning & Bandits

Application Category

• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

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.

Problem

• online learning
• interacting particle systems
• parameter estimation
• stochastic approximation
• asymptotic log-likelihood

Innovation

• online learning
• interacting particle systems
• stochastic approximation
• asymptotic log-likelihood
• parameter estimation