Efficient Online Learning in Interacting Particle Systems

TL;DR

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, is introduced, suggesting that the estimator is effective even in cases where the assumptions required for the theoretical analysis do not hold.

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.

Figures (11)

• Figure 1: Online parameter estimation for a model with quadratic confinement potential and quadratic interaction potential. We plot the sequence of online parameter estimates ${(\bar{\theta}_t^{i,N})_{t\geq 0}}$ and ${(\theta_t^{i,j,k,N})_{t\geq 0}}$, as defined by the update equations in \ref{['eq:IPS_linear_update1']} and \ref{['eq:IPS_linear_update2']}. The true parameters are given by ${\theta_0 = (1.0, 0.2)^{\top}}$. The initial parameter estimates are given by ${\theta_{\mathrm{init},1}\sim \mathcal{U}[1.5,2.5]}$ and ${\theta_{\mathrm{init},2}\sim\mathcal{U}[0.5,1.0]}$.
• Figure 2: The $\mathrm{L}^2$ error of the averaged and the non-averaged estimators, for a model with quadratic confinement potential and quadratic interaction potential. We plot the $\mathrm{L}^2$ error for both estimators after $T=50,000$ iterations, for $N\in\{3,5,10,25,50\}$ particles.
• Figure 3: The asymptotic pseudo log-likelihood function $\mathcal{L}^{i,N}$ for a model with quadratic confinement potential and quadratic interaction potential. We plot the time-averaged likelihood function of the IPS for $N\in\{3,10,20\}$.
• Figure 4: The asymptotic pseudo log-likelihood function $\mathcal{L}^{i,j,k,N}$ for a model with quadratic confinement potential and quadratic interaction potential. We plot the time-averaged likelihood function of the IPS for $N\in\{3,10,20\}$.
• Figure 5: Online parameter estimation for a model with double-well confinement potential and quadratic interaction potential. We plot the sequence of online parameter estimates, as defined by the update equations in \ref{['eq:IPS_bistable_update1']} and \ref{['eq:IPS_bistable_update2']}. The true parameters (black, dashed) are given by ${\theta_0 = (1.0, 2.0, 2.0)^{\top}}$, with the third of these parameters assumed known. The noise coefficient is given by $\sigma=1.0$. The initial parameter estimates are given by ${\theta_{\mathrm{init},1}\sim \mathcal{U}[0.1,0.6]}$ and ${\theta_{\mathrm{init},2}\sim\mathcal{U}[3.0,4.0]}$.

Theorems & Definitions (92)

• Remark 1
• Remark 4
• Remark 6
• Proposition 7
• proof
• Proposition 8
• proof
• Corollary 9
• proof
• Proposition 10