Learning Ergodic Dynamical Systems from a Finite Trajectory

2026-07-24

AI Summary

This work addresses the statistical learning challenges posed by non-i.i.d. data arising from a single finite trajectory of an ergodic stochastic dynamical system, focusing on one-step-ahead prediction modeling. By employing nonlinear least squares to estimate the predictive function and leveraging the invariant measure and uniform geometric ergodicity of the underlying Markov process, the study establishes the first high-probability generalization error bound for non-i.i.d. trajectories grounded in the system’s invariant measure. The approach integrates concentration inequalities for Hilbert space-valued additive functionals with Koopman operator approximation. The theoretical guarantees apply broadly across settings including higher-order systems and finite state spaces, thereby significantly extending the applicability of statistical learning theory to dynamical systems.

Abstract

We consider the problem of learning from a single finite trajectory of an ergodic stochastic dynamical system. More precisely, we study discrete-time autonomous stochastic systems defining time-homogeneous Markov processes. We first focus on estimating the optimal one-step prediction function by nonlinear least squares, and derive high-probability guarantees measured with respect to the invariant measure of the process. These results make explicit how the non-independent and non-identically distributed nature of trajectory data modifies the classical statistical learning analysis. We then extend the framework to higher-order systems and finite-state spaces. Finally, we show that the same least squares and concentration arguments naturally extend to learning Koopman operators. Our approach combines tools from statistical learning theory and quantitative ergodic theory for Markov chains. It relies, in particular, on a concentration inequality for Hilbert-space-valued additive functionals of uniformly geometrically ergodic Markov chains.

Problem

• ergodic dynamical systems
• finite trajectory
• non-iid data
• Markov processes
• statistical learning

Innovation

• ergodic dynamical systems
• non-iid learning
• Koopman operator
• concentration inequality
• statistical learning theory