Learning Ergodic Dynamical Systems from a Finite Trajectory

2026-07-27 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, extended

Summary

This paper introduces a method for learning ergodic stochastic dynamical systems from a single finite trajectory, addressing challenges posed by non-independent and non-identically distributed data. It employs nonlinear least squares estimators to derive high-probability guarantees for one-step prediction functions, measured against the process's invariant measure. The framework is extended to handle vector-valued states, higher-order Markov processes (up to order p), and finite-state spaces, including connections to language models and multiclass classification. A novel concentration inequality for Hilbert-space-valued additive functionals of uniformly geometrically ergodic Markov chains is a core technical contribution. The approach also applies to learning Koopman operators. Numerical results demonstrate that while fast-mixing systems show little difference from i.i.d. learning, slow-mixing bimodal distributions lead to delayed and unstable convergence.

Key takeaway

For Machine Learning Engineers developing models for stochastic dynamical systems, you should account for the non-i.i.d. nature of trajectory data. When using nonlinear least squares, ensure your statistical guarantees explicitly incorporate uniform geometric ergodicity constants, especially for slow-mixing systems. This approach provides robust error bounds for one-step prediction and Koopman operator learning, even from non-stationary initial distributions. Consider the impact of mixing speed on convergence and potential biases.

Key insights

Learning ergodic systems from single trajectories requires specialized statistical guarantees beyond i.i.d. assumptions.

Principles

• Uniform geometric ergodicity enables non-asymptotic guarantees.
• Population risk is defined with respect to the invariant measure.
• Martingale decomposition aids concentration analysis for dependent data.

Method

Uses regularized nonlinear least squares with feature maps to estimate one-step prediction functions or Koopman operators, leveraging a novel Hilbert-space-valued concentration inequality.

In practice

• Apply to vector-valued or higher-order Markov processes.
• Use for next-token prediction in simple language models.
• Evaluate system dynamics via Koopman operator learning.

Topics

• Ergodic Dynamical Systems
• Markov Chains
• Statistical Learning Theory
• Nonlinear Least Squares
• Koopman Operators
• Time Series Forecasting

Best for: Research Scientist, AI Scientist, Machine Learning Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.