We consider distributed optimization problems where networked nodes cooperatively minimize the sum of their locally known convex costs. A popular class of methods to solve these problems are the distributed gradient methods, which are attractive due to their inexpensive iterations, but have a drawback of slow convergence rates. This motivates the incorporation of second order information in the distributed methods, but this task is challenging: although the Hessians which arise in the algorithm design respect the sparsity of the network, their inverses are dense, hence rendering distributed implementations difficult. We overcome this challenge and propose a class of distributed Newton-like methods, which we refer to as Distributed Quasi Newton (DQN). The DQN family approximates the Hessian inverse by: 1) splitting the Hessian into its diagonal and off-diagonal part, 2) inverting the diagonal part, and 3) approximating the inverse of the off-diagonal part through a weighted linear function. The approximation is parameterized by the tuning variables which correspond to different splittings of the Hessian and by different weightings of the off-diagonal Hessian part. Specific choices of the tuning variables give rise to different variants of the proposed general DQN method -dubbed DQN-0, DQN-1 and DQN-2 -which mutually trade-off communication and computational costs for convergence. Simulations demonstrate the effectiveness of the proposed DQN methods.Here, J logis (·) is the logistic loss J logis (z) = log(1 + e −z ), and τ is a positive regularization parameter. Note that, in this example, we have

Abstract-We consider distributed optimization where N nodes in a connected network minimize the sum of their local costs subject to a common constraint set. We propose a distributed projected gradient method where each node, at each iteration k, performs an update (is active) with probability p k , and stays idle (is inactive) with probability 1 − p k . Whenever active, each node performs an update by weight-averaging its solution estimate with the estimates of its active neighbors, taking a negative gradient step with respect to its local cost, and performing a projection onto the constraint set; inactive nodes perform no updates. Assuming that nodes' local costs are strongly convex, with Lipschitz continuous gradients, we show that, as long as activation probability p k grows to one asymptotically, our algorithm converges in the mean square sense (MSS) to the same solution as the standard distributed gradient method, i.e., as if all the nodes were active at all iterations. Moreover, when p k grows to one linearly, with an appropriately set convergence factor, the algorithm has a linear MSS convergence, with practically the same factor as the standard distributed gradient method. Simulations on both synthetic and real world data sets demonstrate that, when compared with the standard distributed gradient method, the proposed algorithm significantly reduces the overall number of per-node communications and per-node gradient evaluations (computational cost) for the same required accuracy.

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Abstract. This paper proposes a new Newton-like method which defines new iterates using a linear system with the same coefficient matrix in each iterate, while the correction is performed on the right-hand-side vector of the Newton system. In this way a method is obtained which is less costly than the Newton method and faster than the fixed Newton method. Local convergence is proved for nonsingular systems. The influence of the relaxation parameter is analyzed and explicit formulae for the selection of an optimal parameter are presented. Relevant numerical examples are used to demonstrate the advantages of the proposed method.