Yuki AKIYAMA

Minh Vu, Yuki Akiyama, Konstantinos Slavakis

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 distribution 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 always the ``optimal'' p-norm for the outlier scenario at hand, outperforming at the same time several non-RL and KBRL schemes.

Yuki Akiyama, Minh Vu, Konstantinos Slavakis

This paper introduces a solution to the problem of selecting dynamically (online) the ``optimal'' p-norm to combat outliers in linear adaptive filtering without any knowledge on the probability density function of the outliers. The proposed online and data-driven framework is built on kernel-based reinforcement learning (KBRL). To this end, novel Bellman mappings on reproducing kernel Hilbert spaces (RKHSs) are introduced. These mappings do not require any knowledge on transition probabilities of Markov decision processes, and are nonexpansive with respect to the underlying Hilbertian norm. The fixed-point sets of the proposed Bellman mappings are utilized to build an approximate policy-iteration (API) framework for the problem at hand. To address the ``curse of dimensionality'' in RKHSs, random Fourier features are utilized to bound the computational complexity of the API. Numerical tests on synthetic data for several outlier scenarios demonstrate the superior performance of the proposed API framework over several non-RL and KBRL schemes.

Yuki Akiyama, Konstantinos Slavakis

This paper introduces novel Bellman mappings (B-Maps) for value iteration (VI) in distributed reinforcement learning (DRL), where agents are deployed over an undirected, connected graph/network with arbitrary topology -- but without a centralized node, that is, a node capable of aggregating all data and performing computations. Each agent constructs a nonparametric B-Map from its private data, operating on Q-functions represented in a reproducing kernel Hilbert space, with flexibility in choosing the basis for their representation. Agents exchange their Q-function estimates only with direct neighbors, and unlike existing DRL approaches that restrict communication to Q-functions, the proposed framework also enables the transmission of basis information in the form of covariance matrices, thereby conveying additional structural details. Linear convergence rates are established for both Q-function and covariance-matrix estimates toward their consensus values, regardless of the network topology, with optimal learning rates determined by the ratio of the smallest positive eigenvalue (the graph's Fiedler value) to the largest eigenvalue of the graph Laplacian matrix. A detailed performance analysis further shows that the proposed DRL framework effectively approximates the performance of a centralized node, had such a node existed. Numerical tests on two benchmark control problems confirm the effectiveness of the proposed nonparametric B-Maps relative to prior methods. Notably, the tests reveal a counter-intuitive outcome: although the framework involves richer information exchange -- specifically through transmitting covariance matrices as basis information -- it achieves the desired performance at a lower cumulative communication cost than existing DRL schemes, underscoring the critical role of sharing basis information in accelerating the learning process.