Learning nonlinear dynamical systems from a single trajectory

• URL: http://arxiv.org/abs/2004.14681v1
• Date: Thu, 30 Apr 2020 10:42:48 GMT
• Title: Learning nonlinear dynamical systems from a single trajectory
• Authors: Dylan J. Foster, Alexander Rakhlin, Tuhin Sarkar
• Abstract summary: We introduce algorithms for learning nonlinear dynamical systems of the form $x_t+1=sigma(Thetastarx_t)+varepsilon_t$. We give an algorithm that recovers the weight matrix $Thetastar$ from a single trajectory with optimal sample complexity and linear running time.
• Score: 102.60042167341956
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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