Single Trajectory Nonparametric Learning of Nonlinear Dynamics

02/16/2022
by   Ingvar Ziemann, et al.
2

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 δ^-q. Here, T is the length of the observed trajectory, δ∈ℝ_+ is the packing granularity and q∈ (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).

READ FULL TEXT

page 1

page 2

page 3

page 4

07/29/2022

Ensemble forecasts in reproducing kernel Hilbert space family: dynamical systems in Wonderland

A methodological framework for ensemble-based estimation and simulation ...
12/07/2021

Bless and curse of smoothness and phase transitions in nonparametric regressions: a nonasymptotic perspective

When the regression function belongs to the standard smooth classes cons...
05/31/2018

Metric on Nonlinear Dynamical Systems with Koopman Operators

The development of a metric for structural data is a long-term problem i...
02/06/2021

Efficient Learning of a Linear Dynamical System with Stability Guarantees

We propose a principled method for projecting an arbitrary square matrix...
05/17/2019

Modeling of Missing Dynamical Systems: Deriving Parametric Models using a Nonparametric Framework

In this paper, we consider modeling missing dynamics with a non-Markovia...
08/13/2020

Learning Stability Certificates from Data

Many existing tools in nonlinear control theory for establishing stabili...
07/29/2019

Learning Stabilizable Nonlinear Dynamics with Contraction-Based Regularization

We propose a novel framework for learning stabilizable nonlinear dynamic...