[edit]
Learning nonlinear dynamical systems from a single trajectory
Proceedings of the 2nd Conference on Learning for Dynamics and Control, PMLR 120:851-861, 2020.
Abstract
We introduce algorithms for learning nonlinear dynamical systems of theform xt+1=σ(Θxt)+εt, where Θ is a weightmatrix, σ is a nonlinear monotonic link function, andεt is a mean-zero noise process. When the link function is known, wegive an algorithm that recovers the weight matrix Θ from a single trajectorywith optimal sample complexity and linear running time. The algorithmsucceeds under weaker statistical assumptions than in previous work, and inparticular i) does not require a bound on the spectral norm of the weightmatrix Θ (rather, it depends on a generalization of thespectral radius) and ii) works when the link function is the ReLU. Our analysis has three keycomponents: i) We show how \emph{sequential Rademacher complexities} can beused to provide generalization guarantees for general dynamicalsystems, ii) we give a general recipe whereby global stability fornonlinear dynamical systems can be used to certify that the state-vector covariance is well-conditioned, and iii) using these tools, we extend well-known algorithms for efficiently learning generalized linear models to the dependent setting.