Distributed Newton Method Over Graphs: Can Sharing of Second-Order Information Eliminate the Condition Number Dependence?

One of the main advantages of second-order methods in a centralized setting is that they are insensitive to the condition number of the objective function's Hessian. For applications such as regression analysis, this means that less pre-processing of the data is required for the algorithm to work well, as the ill-conditioning caused by highly correlated variables will not be as problematic. Similar condition number independence has not yet been established for distributed methods. In this paper, we analyze the performance of a simple distributed second-order algorithm on quadratic problems and show that its convergence depends only logarithmically on the condition number. Our empirical results indicate that the use of second-order information can yield large efficiency improvements over first-order methods, both in terms of iterations and communications, when the condition number is of the same order of magnitude as the problem dimension.