Reliable sampling-based RKHS norm estimation via superconvergence
This removes a key obstacle to deploying learning-based control algorithms that rely on RKHS norm-based error bounds.
The authors propose a sampling-based method to estimate the RKHS norm of a target function, which is crucial for error bounds in kernel methods but typically unknown. Their approach, grounded in superconvergence theory, works for broad function classes and is validated by numerical experiments.
Kernel methods are one of the cornerstones of learning-based control, modern system identification, surrogate modelling, and related fields. A key advantage of this class of learning and function approximation methods is the availability of quantitative error bounds, which in turn play a key role in guaranteeing the safety of learned controllers and related learning-based algorithms. However, these error bounds rely on a particular property of the target function -- its reproducing kernel Hilbert space (RKHS) norm -- which is usually impossible to obtain in practice. Motivated by this severe shortcoming, we present a novel sampling-based RKHS norm estimation approach with a solid theoretical foundation, leveraging very recent advances in the theory of superconvergence in kernel methods. Our method is applicable to a broad range of practically relevant function classes and requires only reasonable prior knowledge about the target function. Extensive numerical experiments demonstrate the efficacy and practical applicability of the proposed method. By providing a reliable RKHS norm estimation approach, we remove a major obstacle to the practical deployment of learning-based control algorithms.