Nonparametric Sparse Online Learning of the Koopman Operator

2024-05-13

arXiv.org

AI Summary

This work addresses modeling bias in Koopman operator learning for nonlinear dynamical systems under non-closed function spaces. Methodologically, it introduces a nonparametric sparse online learning framework, formulating the Koopman operator action as a conditional mean embedding (CME) in a reproducing kernel Hilbert space (RKHS) and devising a trajectory-sampling-based online sparse stochastic approximation algorithm. Theoretical contributions include the first convergence analysis for online Koopman learning in non-closed, nonparametric settings, yielding a finite-time last-iterate error bound and extending beyond classical finite-dimensional stochastic approximation frameworks. Experiments demonstrate substantial improvements in generalization performance and computational efficiency over existing approaches.

Technology Category

• Machine Learning: Online Learning & Bandits
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Social networks and social learning
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The Koopman operator provides a powerful framework for representing the dynamics of general nonlinear dynamical systems. Data-driven techniques to learn the Koopman operator typically assume that the chosen function space is closed under system dynamics. In this paper, we study the Koopman operator via its action on the reproducing kernel Hilbert space (RKHS), and explore the mis-specified scenario where the dynamics may escape the chosen function space. We relate the Koopman operator to the conditional mean embeddings (CME) operator and then present an operator stochastic approximation algorithm to learn the Koopman operator iteratively with control over the complexity of the representation. We provide both asymptotic and finite-time last-iterate guarantees of the online sparse learning algorithm with trajectory-based sampling with an analysis that is substantially more involved than that for finite-dimensional stochastic approximation. Numerical examples confirm the effectiveness of the proposed algorithm.

Problem

• Learning Koopman operator dynamics
• Handling mis-specified function spaces
• Online sparse learning algorithm

Innovation

• Nonparametric sparse online learning
• Reproducing kernel Hilbert space
• Operator stochastic approximation algorithm