Nonparametric Bellman Mappings for Reinforcement Learning: Application to Robust Adaptive Filtering

This paper designs novel nonparametric Bellman mappings in reproducing kernel Hilbert spaces (RKHSs) for reinforcement learning (RL). The proposed mappings benefit from the rich approximating properties of RKHSs, adopt no assumptions on the statistics of the data owing to their nonparametric nature, require no knowledge on transition probabilities of Markov decision processes, and may operate without any training data. Moreover, they allow for sampling on-the-fly via the design of trajectory samples, re-use past test data via experience replay, effect dimensionality reduction by random Fourier features, and enable computationally lightweight operations to fit into efficient online or time-adaptive learning. The paper offers also a variational framework to design the free parameters of the proposed Bellman mappings, and shows that appropriate choices of those parameters yield several popular Bellman-mapping designs. As an application, the proposed mappings are employed to offer a novel solution to the problem of countering outliers in adaptive filtering. More specifically, with no prior information on the statistics of the outliers and no training data, a policy-iteration algorithm is introduced to select online, per time instance, the “optimal” coefficient $p$ in the least-mean-$p$-power-error method. Numerical tests on synthetic data showcase, in most of the cases, the superior performance of the proposed solution over several RL and non-RL schemes.