Near-optimal Offline and Streaming Algorithms for Learning Non-Linear Dynamical Systems

Abstract

We consider the setting of vector valued non-linear dynamical systems X t+1 = φ(A * X t ) + η t , where η t is unbiased noise and φ : R → R is a known link function that satisfies certain expansivity property. The goal is to learn A * from a single trajectory X 1 , • • • , X T of dependent or correlated samples. While the problem is well-studied in the linear case, where φ is identity, with optimal error rates even for non-mixing systems, existing results in the non-linear case hold only for mixing systems. In this work, we improve existing results for learning nonlinear systems in a number of ways: a) we provide the first offline algorithm that can learn non-linear dynamical systems without the mixing assumption, b) we significantly improve upon the sample complexity of existing results for mixing systems, c) in the much harder one-pass, streaming setting we study a SGD with Reverse Experience Replay (SGD − RER) method, and demonstrate that for mixing systems, it achieves the same sample complexity as our offline algorithm, d) we justify the expansivity assumption by showing that for the popular ReLU link function -a non-expansive but easy to learn link function with i.i.d. samples -any method would require exponentially many samples (with respect to dimension of X t ) from the dynamical system. We validate our results via. simulations and demonstrate that a naive application of SGD can be highly sub-optimal. Indeed, our work demonstrates that for correlated data, specialized methods designed for the dependency structure in data can significantly outperform standard SGD based methods.Preprint. Under review.

Single Trajectory Nonparametric Learning of Nonlinear Dynamics

“…By contrast, previous work assumes i.i.d. data or focuses on either linear models [Simchowitz et al, 2018, Tsiamis and Pappas, 2019, Jedra and Proutiere, 2020 or parametric models with known nonlinearities , Sattar and Oymak, 2020, Jain et al, 2021.…”

“…In contrast, it is well-known that this dependency can be avoided for learning in linear systems [Lai andWei, 1982, Simchowitz et al, 2018]. Recently Jain et al [2021] showed under a strong invertibility condition that dependency on the mixing time can also be avoided for the generalized linear model ( 16). This leaves open the question whether learning without mixing is possible in situations beyond the generalized linear model.…”

“…The same rate can also be achieved for linear systems with more general input-output behavior Ozay, 2019, Tsiamis andPappas, 2019]. Moving to nonlinear models, recursive and gradient type algorithms can be shown to converge at a rate of T −1/2 for the generalized linear model f (x t ) = φ(A x t ), where φ is a known Lipschitz link function , Sattar and Oymak, 2020, Jain et al, 2021. In this paper, we significantly generalize these results and provide rate-optimal error bounds for nonparametric function classes in terms of their metric entropies.…”

“…The next example revisits the generalized linear models using Theorem 3. This model has recently been analyzed using recursive methods , Sattar and Oymak, 2020, Jain et al, 2021.…”