Koopman Kernel Regression
This addresses the problem of unreliable forecasts in machine learning for decision-making, such as in reinforcement learning, by providing a method with learning-theoretic guarantees, though it appears incremental as it builds on Koopman operator theory with new convergence results.
The paper tackles the challenge of forecasting complex nonlinear dynamical systems for decision-making by introducing Koopman Kernel Regression (KKR), which transforms forecasts into linear time-invariant ODEs using a universal Koopman-invariant RKHS, resulting in superior forecasting performance compared to existing methods in experiments.
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.