Improved Convergence Rates of P-EXTRA for Non-smooth Distributed optimization

P-EXTRA is a powerful distributed algorithm for nonsmooth, convex optimization over networks, which allows nodes in a network to cooperatively reach a consensus and meanwhile minimize the sum of their individual cost functions. Nevertheless, only convergence rates in terms of an optimality residual have been provided for P-EXTRA so far. In this paper, we show that the objective function value at the running average of the iterates generated by P-EXTRA converges to the optimal value at an O(1/k) rate, which is a new convergence rate result for P-EXTRA. We also significantly improve the known o(1/k) rate of the consensus error for P-EXTRA to O (1/k2). All these results are established under a more general parameter condition and through completely different convergence analysis, compared with the existing work. Finally, we demonstrate our results via numerical examples.