Learning Ergodic Dynamical Systems from a Finite Trajectory
Oleksii Kachaiev, Silvia Villa, Lorenzo Rosasco
stat.ML, cs.LG
Submitted: 2026-07-24
License: http://creativecommons.org/licenses/by/4.0/
The gist: We consider the problem of learning from a single finite trajectory of an ergodic stochastic dynamical system.
Terminology
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.
Sources
- Modern Koopman Theory for Dynamical Systems
- Hoeffding's Inequality for Markov Chains under Generalized Concentrability Condition
- An Introductory Guide to Koopman Learning
- Modelling transition dynamics in MDPs with RKHS embeddings
- Nonparametric Sparse Online Learning of the Koopman Operator
- Bernstein-type Inequalities for Markov Chains and Markov Processes: A Simple and Robust Proof
- Theory and Algorithms for Forecasting Time Series
- On the Consistency of Kernel Methods with Dependent Observations
- Koopman Operator, Geometry, and Learning
- Online Learning for Nonlinear Dynamical Systems without the I.I.D. Condition
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