Learning nonlinear dynamical systems from a single trajectory

TL;DR

This paper presents an efficient algorithm for learning nonlinear dynamical systems from a single trajectory, requiring weaker assumptions and handling non-strictly-increasing link functions like ReLU.

Contribution

It introduces a novel algorithm that recovers the system's weight matrix with optimal sample complexity, even under weaker statistical conditions and for a broader class of nonlinearities.

Findings

Algorithm recovers weights with optimal sample complexity

Works under weaker assumptions than previous methods

Handles non-strictly-increasing link functions like ReLU

Abstract

We introduce algorithms for learning nonlinear dynamical systems of the form $x_{t+1}=\sigma(\Theta^{\star}x_t)+\varepsilon_t$, where $\Theta^{\star}$ is a weight matrix, $\sigma$ is a nonlinear link function, and $\varepsilon_t$ is a mean-zero noise process. We give an algorithm that recovers the weight matrix $\Theta^{\star}$ from a single trajectory with optimal sample complexity and linear running time. The algorithm succeeds under weaker statistical assumptions than in previous work, and in particular i) does not require a bound on the spectral norm of the weight matrix $\Theta^{\star}$ (rather, it depends on a generalization of the spectral radius) and ii) enjoys guarantees for non-strictly-increasing link functions such as the ReLU. Our analysis has two key components: i) we give a general recipe whereby global stability for nonlinear dynamical systems can be used to certify that the state-vector covariance is well-conditioned, and ii) using these tools, we extend well-known algorithms for efficiently learning generalized linear models to the dependent setting.