We propose a communication and computation efficient second-order method for distributed optimization. For each iteration, our method only requires $\mathcal{ O}(d)$ communication complexity, where $d$ is the problem dimension. We also provide theoretical analysis to show the proposed method has the similar convergence rate as the classical second-order optimization algorithms. Concretely, our method can find~$\big(\epsilon, \sqrt{ dL\epsilon}\,\big)$-second-order stationary points for nonconvex problem by $\mathcal{ O}\big(\sqrt{ dL}\,\epsilon^{ -3/2}\big)$ iterations, where $L$ is the Lipschitz constant of Hessian. Moreover, it enjoys a local superlinear convergence under the strongly-convex assumption. Experiments on both convex and nonconvex problems show that our proposed method performs significantly better than baselines.
Categories: math.OC