Sparsity-based regularization for learning over multitask graphs

In this work, we consider learning over multitask graphs, where each agent seeks to estimate its own parameter vector (task) while benefiting from collaboration with its neighboring agents with related tasks. A key method for promoting relationships between tasks is regularization. Unlike previous studies that enforce graph smoothness via graph-based smooth regularization, we investigate non-smooth regularization to promote sparsity and piecewise-constant transitions between tasks at neighboring agents. We formulate a globally regularized optimization problem and, motivated by the forward-backward splitting approach, we propose a decentralized approach for solving the non-smooth optimization problem. Then, under some general convexity assumptions on the individual costs and regularization, we study the convergence w.r.t. the optimal solution of the globally regularized cost, in the mean-square-error sense. Finally, we illustrate the theoretical findings and the effectiveness of the proposed approach through simulations.