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Newton-like Method with Diagonal Correction for Distributed Optimization

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.