Koopman Kernel Regression
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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 dynamical systems. This makes system analysis and long-term predictions simple – involving only matrix multiplications. However, the transformation to a linear system is generally non-trivial and unknown, requiring learning-based approaches. While there exists a variety of approaches, they usually lack crucial learning-theoretic guarantees, such that the behavior of the obtained models with increasing data and dimensionality is often unclear. We address the aforementioned by deriving a novel reproducing kernel Hilbert space (RKHS) that solely spans transformations into linear 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 risk bounds under weaker assumptions than existing work. Our numerical experiments indicate advantages over state-of-the-art statistical learning approaches for Koopman-based predictors.
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