Single Trajectory Nonparametric Learning of Nonlinear Dynamics

• URL: http://arxiv.org/abs/2202.08311v1
• Date: Wed, 16 Feb 2022 19:38:54 GMT
• Title: Single Trajectory Nonparametric Learning of Nonlinear Dynamics
• Authors: Ingvar Ziemann, Henrik Sandberg, Nikolai Matni
• Abstract summary: Given a single trajectory of a dynamical system, we analyze the performance of the nonparametric least squares estimator (LSE) We leverage recently developed information-theoretic methods to establish the optimality of the LSE for non hypotheses classes. We specialize our results to a number of scenarios of practical interest, such as Lipschitz dynamics, generalized linear models, and dynamics described by functions in certain classes of Reproducing Kernel Hilbert Spaces (RKHS)
• Score: 8.438421942654292
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Given a single trajectory of a dynamical system, we analyze the performance of the nonparametric least squares estimator (LSE). More precisely, we give nonasymptotic expected $l^2$-distance bounds between the LSE and the true regression function, where expectation is evaluated on a fresh, counterfactual, trajectory. We leverage recently developed information-theoretic methods to establish the optimality of the LSE for nonparametric hypotheses classes in terms of supremum norm metric entropy and a subgaussian parameter. Next, we relate this subgaussian parameter to the stability of the underlying process using notions from dynamical systems theory. When combined, these developments lead to rate-optimal error bounds that scale as $T^{-1/(2+q)}$ for suitably stable processes and hypothesis classes with metric entropy growth of order $\delta^{-q}$. Here, $T$ is the length of the observed trajectory, $\delta \in \mathbb{R}_+$ is the packing granularity and $q\in (0,2)$ is a complexity term. Finally, we specialize our results to a number of scenarios of practical interest, such as Lipschitz dynamics, generalized linear models, and dynamics described by functions in certain classes of Reproducing Kernel Hilbert Spaces (RKHS).

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