The chapter shows how the classical adaptive filtering algorithms can be adapted to distributed learning. In distributed learning, there is a set of adaptive filtering placed at nodes utilizing a local input and desired signals. These distributed networks of sensor nodes are located at distinct positions, which might improve the reliability and robustness of the parameter estimation in comparison to stand-alone adaptive filters. In distributed adaptive networks, parameter estimation might be obtained in a centralized form or a decentralized form. The centralized case processes the signals from all nodes of the network in a single fusion center, whereas in the decentralized case, processing is performed locally followed by a proper combination of partial estimates to result in a consensus parameter estimate. The main drawbacks of the centralized configuration are its data communication and computational costs, particularly in networks with a large number of nodes. On the other hand, the decentralized estimators require fewer data to feed the estimators and improve on robustness. The provides a discussion on equilibrium and consensus using arguments drawn from the pari-mutuel betting system. The expert opinion pool is the concept to induce improved estimation and data modeling, utilizing De-Groot’s algorithm and Markov chains as tools to probate equilibrium at consensus. It also introduces the distributed versions of the LMS and RLS adaptive filtering algorithms with emphasis on the decentralized parameter estimation case. This chapter also addresses how data broadcasting can be confined to a subset of nodes so that the overall network reduces the power consumption and bandwidth usage. Then, the chapter discusses a strategy to incorporate a data selection based on the SM adaptive filtering.

Chapter 2 presents several strategies to exploit sparsity in the parameters being estimated in order to obtain better estimates and accelerate convergence, two advantages of paramount importance when dealing with real problems requiring the estimation of many parameters. In these cases, the classical adaptive filtering algorithms exhibit a slow and often unacceptable convergence rate. In this chapter, many algorithms capable of exploiting sparse models are presented. Also, the two most widely used approaches to exploit sparsity are presented, and their pros and cons are discussed. The first approach explicitly models sparsity by relying on sparsity-promoting regularization functions. The second approach utilizes updates proportional to the magnitude of the coefficient being updated, thus accelerating the convergence of large magnitude coefficients. After reading this chapter, the reader will not only obtain a deeper understanding of the subject but also be able to adapt or develop algorithms based on his own needs.

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