Online Proximal Learning Over Jointly Sparse Multitask Networks With $\ell _{\infty, 1}$ Regularization

Abstract

Modeling relations between local optimum parameter vectors to estimate in multitask networks has attracted much attention over the last years. This work considers a distributed optimization problem with jointly sparse structure among nodes, that is, the local solutions have the same sparse support set. Several mixed norm have been proposed to address the jointly sparse structure in the literature. Among several candidates, the (reweighted) $\ell _{\infty,1}$ -norm is element-wise separable, it is more convenient to evaluate their approximate proximal operators. Thus by introducing a (reweighted) $\ell _{\infty,1}$ -norm penalty term at each node, and using a proximal gradient method to minimize the regularized cost, we devise a proximal multitask diffusion LMS algorithm which can promote joint-sparsity. Analyses are provided to characterize the algorithm behavior in the mean and mean-square sense. Simulation results are presented to show its effectiveness, as well as the accuracy of the theoretical findings.