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We propose a distributed cubic regularization of the Newton method forsolving (constrained) empirical risk minimization problems over a network ofagents, modeled as undirected graph. The algorithm employs an inexact,preconditioned Newton step at each agent's side: the gradient of thecentralized loss is iteratively estimated via a gradient-tracking consensusmechanism and the Hessian is subsampled over the local data sets. No Hessianmatrices are thus exchanged over the network. We derive global complexitybounds for convex and strongly convex losses. Our analysis reveals aninteresting interplay between sample and iteration/communication complexity:statistically accurate solutions are achievable in roughly the same number ofiterations of the centralized cubic Newton method, with a communication costper iteration of the order of$\widetilde{\mathcal{O}}\big(1/\sqrt{1-\rho}\big)$, where $\rho$ characterizesthe connectivity of the network. This demonstrates a significant communicationsaving with respect to that of existing, statistically oblivious, distributedNewton-based methods over networks.

Newton Method over Networks is Fast up to the Statistical Precision