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This research advances the theoretical foundations for learning stochastic dynamical systems from limited trajectory data using nonlinear least squares and Koopman operator methods. By deriving high-probability guarantees that account for non-i.i.d. dependencies in Markov chains, it provides rigorous tools applicable to complex AI models where standard statistical assumptions fail. The work bridges quantitative ergodic theory with modern machine learning frameworks to improve prediction accuracy under data constraints.
Learning Ergodic Dynamical Systems from a Finite Trajectory
arXiv:2607.22399v1 Announce Type: cross 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.
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.
Machine Learning (stat.ML); Machine Learning (cs.LG)
arXiv:2607.22399 [stat.ML]
(or arXiv:2607.22399v1 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2607.22399 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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