  We consider a distributed learning setup where a sparse signal is estimated
over a network. Our main interest is to save communication resource for
information exchange over the network and reduce processing time. Each node of
the network uses a convex optimization based algorithm that provides a locally
optimum solution for that node. The nodes exchange their signal estimates over
the network in order to refine their local estimates. At a node, the
optimization algorithm is based on an $\ell_1$-norm minimization with
appropriate modifications to promote sparsity as well as to include influence
of estimates from neighboring nodes. Our expectation is that local estimates in
each node improve fast and converge, resulting in a limited demand for
communication of estimates between nodes and reducing the processing time. We
provide restricted-isometry-property (RIP)-based theoretical analysis on
estimation quality. In the scenario of clean observation, it is shown that the
local estimates converge to the exact sparse signal under certain technical
conditions. Simulation results show that the proposed algorithms show
competitive performance compared to a globally optimum distributed LASSO
algorithm in the sense of convergence speed and estimation error.
