Non-asymptotic and Accurate Learning of Nonlinear Dynamical Systems

Abstract

We consider the problem of learning stabilizable systems governed by nonlinear state equation $h_{t+1}=\varphi (h_t,u_t;\theta )+w_t$. 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 temporally-dependent samples can be approximated by i.i.d. samples via a truncation argument by using mixing-time 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 ground-truth 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 entry-wise nonlinear activations and verify our theory in various numerical experiments

presentation improved, proof sketch added, Assumption 2(b) removed, references added

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• Efficient Learning of Generalized Linear and Single Index Models with Isotonic Regression
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• Streaming Linear System Identification with Reverse Experience Replay
• Exact Asymptotics for Linear Quadratic Adaptive Control
• Modeling Latent Non-Linear Dynamical System over Time Series
• Online Stochastic Gradient Descent Learns Linear Dynamical Systems from A Single Trajectory
• Near-optimal Offline and Streaming Algorithms for Learning Non-Linear Dynamical Systems
• Learning the Linear Quadratic Regulator from Nonlinear Observations
• Identification and Adaptive Control of Markov Jump Systems: Sample Complexity and Regret Bounds