Optimal Diffusion Learning Over Networks—Part II: Multitask Algorithms

Abstract

In Part I of this presentation, we have formulated single and multitask quadratic optimization problems, where agents are subject to quadratic, smoothing constraints over a graph. We have focused particularly on single task designs, whereby node uncertainties and their strength relative to un-regularized cost are tackled altogether by means of an adaptive penalty function. In this sequel, we readdress the multitask problem and propose new distributed implementations for their corresponding exact <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">leaky</i> -RLS solutions. We motivate a network formulation from a standalone viewpoint by capitalizing on the fact that 1) for regressors having uncorrelated entries, the performance of an efficient <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\mathcal O}(M^2)$</tex-math></inline-formula> conjugate-gradient (CG) realization of the leaky LS solution is identical to the one of an RLS filter; 2) a CG implementation does not require inversion of the underlying sample covariance matrix. Simple arguments yield an extended network-CG algorithm that relies on node-level recursions employing distinct step-sizes. Unlike the exponentially-weighted RLS algorithm, which tapers off regularization over time, a persistent penalty strength conforms with the very purpose of the equivalent network trust-region problem, while granting a well-conditioned solution. The approach further yields another family of single-task algorithms in terms of network linearly constrained solutions, which can be contrasted to the ones proposed in Part I. In particular, the exact linearly-constrained network LMS implementation proposed here outperforms the adaptive relative-variance NLMS, under much lower computational requirements.