Decentralized Approximate Newton Methods for Convex Optimization on Networked Systems

• Published in: IEEE transactions on control of network systems Vol. 8; no. 3; pp. 1489 - 1500
• Main Authors: Wei, Hejie, Qu, Zhihai, Wu, Xuyang, Wang, Hao, Lu, Jie
• Format: Journal Article
• Language: English
• Published: Piscataway IEEE 01.09.2021; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Approximation; Approximation algorithms; Computational geometry; Convergence; Convex analysis; Convex functions; Convexity; Distributed optimization; Linear programming; network optimization; Newton method; Newton methods; Nodes; Optimization; Radio frequency; second-order methods
• ISSN: 2325-5870; 2372-2533
• DOI: 10.1109/TCNS.2021.3070663

In this article, a class of decentralized approximate Newton (DEAN) methods for addressing convex optimization on a networked system is developed, where nodes in the networked system seek a consensus that minimizes the sum of their individual objective functions through local interactions only. The proposed DEAN algorithms allow each node to repeatedly perform a local approximate Newton update, which leverages tracking the global Newton direction and dissipating the discrepancies among the nodes. Under less restrictive problem assumptions in comparison with most existing second-order methods, the DEAN algorithms enable the nodes to reach a consensus that can be arbitrarily close to the optimum. Moreover, for a particular DEAN algorithm, the nodes linearly converge to a common suboptimal solution with an explicit error bound. Finally, simulations demonstrate the competitive performance of DEAN in convergence speed, accuracy, and efficiency.