Nonasymptotic and Accurate Learning of Nonlinear Dynamical Systems
Abstract
We consider the problem of learning stabilizable systems governed by nonlinear state equation $h_{t+1}=\phi(h_t,u_t;\theta)+w_t$. Here $\theta$ is the unknown system dynamics, $h_t $ is the state, $u_t$ is the input and $w_t$ is the additive noise vector. We study gradient based algorithms to learn the system dynamics $\theta$ from samples obtained from a single finite trajectory. If the system is run by a stabilizing input policy, we show that temporallydependent samples can be approximated by i.i.d. samples via a truncation argument by using mixingtime arguments. We then develop new guarantees for the uniform convergence of the gradients of empirical loss. Unlike existing work, our bounds are noise sensitive which allows for learning groundtruth dynamics with high accuracy and small sample complexity. Together, our results facilitate efficient learning of the general nonlinear system under stabilizing policy. We specialize our guarantees to entrywise nonlinear activations and verify our theory in various numerical experiments
 Publication:

arXiv eprints
 Pub Date:
 February 2020
 arXiv:
 arXiv:2002.08538
 Bibcode:
 2020arXiv200208538S
 Keywords:

 Computer Science  Machine Learning;
 Electrical Engineering and Systems Science  Systems and Control;
 Mathematics  Optimization and Control;
 Statistics  Applications;
 Statistics  Machine Learning
 EPrint:
 presentation improved, proof sketch added, Assumption 2(b) removed, references added