Dynamic Selection of p-norm in Linear Adaptive Filtering via online Kernel-based Reinforcement Learning

This study addresses the problem of selecting dynamically, at each time instance, the "optimal" p-norm to combat outliers in linear adaptive filtering without any knowledge on the potentially time-varying probability density function of the outliers. To this end, an online and data-driven framework is designed via kernel-based reinforcement learning (KBRL). Novel Bellman mappings on reproducing kernel Hilbert spaces (RKHSs) are introduced that need no knowledge on transition probabilities of Markov decision processes, and are nonexpansive with respect to the underlying Hilbertian norm. An approximate policy-iteration framework is finally offered via the introduction of a finite-dimensional affine superset of the fixed-point set of the proposed Bellman mappings. The well-known "curse of dimensionality" in RKHSs is addressed by building a basis of vectors via an approximate linear dependency criterion. Numerical tests on synthetic data demonstrate that the proposed framework selects the "optimal" p-norm for the outlier scenario at hand at every time instance, outperforming several non-RL and KBRL schemes.