Distributed Adaptive Greedy Quasi-Newton Methods with Explicit Non-asymptotic Convergence Bounds

Abstract

Though quasi-Newton methods have been extensively studied in the literature, they either suffer from local convergence or use a series of line searches for global convergence which is not acceptable in the distributed setting. In this work, we first propose a line search free greedy quasi-Newton (GQN) method with adaptive steps and establish explicit non-asymptotic bounds for both the global convergence rate and local superlinear rate. Our novel idea lies in the design of multiple greedy quasi-Newton updates, which involves computing Hessian-vector products, to control the Hessian approximation error, and a simple mechanism to adjust stepsizes to ensure the objective function improvement per iterate. Then, we extend it to the master-worker framework and propose a distributed adaptive GQN method whose communication cost is comparable with that of first-order methods, yet it retains the superb convergence property of its centralized counterpart. Finally, we demonstrate the advantages of our methods via numerical experiments.