Koopman Kernel Regression

Petar Bevanda, Max Beier, Armin Lederer, Stefan Georg Sosnowski, Eyke Hüllermeier, Sandra Hirche

Keywords: Kernel Methods, Regression, Statistical Learning Theory, Koopman Operator, Mode Decomposition, Dynamical Systems, Supervised Learning

TL;DR: We propose a novel Koopman-invariant and universal kernel on trajectories with generalization and convergence guarantees for LTI forecasting.

Abstract:

Many machine learning approaches for decision making, such as reinforcement learning, rely on simulators or predictive models to forecast the time-evolution of quantities of interest, e.g., the state of an agent or the reward of a policy. Forecasts of such complex phenomena are commonly described by highly nonlinear dynamical systems, making their use in optimization-based decision-making challenging.
Koopman operator theory offers a beneficial paradigm for addressing this problem by characterizing forecasts via linear time-invariant (LTI) ODEs, turning multi-step forecasts into sparse matrix multiplication.
Though there exists a variety of learning approaches, they usually lack crucial learning-theoretic guarantees, making the behavior of the obtained models with increasing data and dimensionality unclear.
We address the aforementioned by deriving a universal Koopman-invariant reproducing kernel Hilbert space (RKHS) that solely spans transformations into LTI dynamical systems. The resulting Koopman Kernel Regression (KKR) framework enables the use of statistical learning tools from function approximation for novel convergence results and generalization error bounds under weaker assumptions than existing work. Our experiments demonstrate superior forecasting performance compared to Koopman operator and sequential data predictors in RKHS.

Submission Number: 5469