Distributed Newton Optimization With Maximized Convergence Rate

The distributed optimization problem is set up in a collection of nodes interconnected via a communication network. The goal is to find the minimizer of a global objective function formed by the sum of local functions known at individual nodes. A number of methods, having different advantages, are available for addressing this problem. The goal of this article is to achieve the maximum possible convergence rate. As the first step toward this end, we propose a new method, which we show converges faster than other available options. As the second step toward our goal, we complement the proposed method with a fully distributed method for estimating the optimal step size that maximizes the convergence rate. We provide theoretical guarantees for the convergence of the resulting method in a neighborhood of the solution. We present numerical experiments showing that, when using the same step size, our method converges significantly faster than its rivals. Experiments also show that the distributed step-size estimation method achieves an asymptotic convergence rate very close to the theoretical maximum.