交互粒子系统中高效的在线学习
Efficient Online Learning in Interacting Particle Systems
AI总结：
本文提出了一种在交互粒子系统中进行高效在线学习的方法，通过连续观测少量粒子并递归更新模型参数，实现了对渐近对数似然的收敛性证明，并通过数值实验验证了其有效性。
摘要：
我们介绍了一种新的方法，用于在随机交互粒子系统中进行在线参数估计，该方法基于对系统中少量粒子的连续观测。我们的方法通过递归更新模型参数，利用渐近对数似然梯度的随机近似来实现，该梯度是通过连续的观测流计算得到的。在合适的假设下，我们严格证明了该方法收敛到交互粒子系统的渐近对数似然的平稳点。我们考虑了时间范围$ t \rightarrow \infty $的极限情况，对于固定且有限数量的粒子，以及粒子数量$ N \rightarrow \infty $和时间范围$ t \rightarrow \infty $的联合极限情况。在对渐近对数似然的额外假设下，我们还建立了$ \mathrm{L}^2 $收敛速度和中心极限定理。最后，我们展示了几个具有实际意义的数值例子，包括系统风险模型、相互作用的FitzHugh-Nagumo神经元模型以及Cucker-Smale flocking模型。我们的数值结果证实了我们的理论结果，并表明即使在理论分析所需假设不成立的情况下，我们的估计器也有效。
英文摘要：
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.
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