Learning nonlinear dynamical systems from a single trajectory

We introduce algorithms for learning nonlinear dynamical systems of the form xt+1=σ(Thetastarxt)+varepsilont, where Thetastar is a weight matrix, σ is a nonlinear link function, and varepsilont is a mean-zero noise process. We give an algorithm that recovers the weight matrix Thetastar 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 Thetastar (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.