Network Newton-Part I: Algorithm and Convergence

We study the problem of minimizing a sum of convex objective functions where the components of the objective are available at different nodes of a network and nodes are allowed to only communicate with their neighbors. The use of distributed gradient methods is a common approach to solve this problem. Their popularity notwithstanding, these methods exhibit slow convergence and a consequent large number of communications between nodes to approach the optimal argument because they rely on first order information only. This paper proposes the network Newton (NN) method as a distributed algorithm that incorporates second order information. This is done via distributed implementation of approximations of a suitably chosen Newton step. The approximations are obtained by truncation of the Newton step's Taylor expansion. This leads to a family of methods defined by the number $K$ of Taylor series terms kept in the approximation. When keeping $K$ terms of the Taylor series, the method is called NN-$K$ and can be implemented through the aggregation of information in $K$-hop neighborhoods. Convergence to a point close to the optimal argument at a rate that is at least linear is proven and the existence of a tradeoff between convergence time and the distance to the optimal argument is shown. Convergence rate, several practical implementation matters, and numerical analyses are presented in a companion paper [3].