Distributed Heavy-Ball: A Generalization and Acceleration of First-Order Methods With Gradient Tracking

• Published in: IEEE transactions on automatic control Vol. 65; no. 6; pp. 2627 - 2633
• Main Authors: Xin, Ran, Khan, Usman A.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.06.2020; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Accelerated first-order methods; Acceleration; Convergence; cooperative control; distributed optimization; Eigenvectors; First order algorithms; Graph theory; Heart; heavy-ball momentum; Linear programming; Momentum; Optimization; Radio frequency; Topology; Tracking
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2019.2942513

We study distributed optimization to minimize a sum of smooth and strongly-convex functions. Recent work on this problem uses gradient tracking to achieve linear convergence to the exact global minimizer. However, a connection among different approaches has been unclear. In this paper, we first show that many of the existing first-order algorithms are related with a simple state transformation, at the heart of which lies a recently introduced algorithm known as <inline-formula><tex-math notation="LaTeX">\mathcal {AB}</tex-math></inline-formula>. We then present distributed heavy-ball , denoted as <inline-formula><tex-math notation="LaTeX">\mathcal {AB}m</tex-math></inline-formula>, that combines <inline-formula><tex-math notation="LaTeX">\mathcal {AB}</tex-math></inline-formula> with a momentum term and uses nonidentical local step-sizes. By simultaneously implementing both row- and column-stochastic weights, <inline-formula><tex-math notation="LaTeX">\mathcal {AB}m</tex-math></inline-formula> removes the conservatism in the related work due to doubly stochastic weights or eigenvector estimation. <inline-formula><tex-math notation="LaTeX">\mathcal {AB}m</tex-math></inline-formula> thus naturally leads to optimization and average consensus over both undirected and directed graphs. We show that <inline-formula><tex-math notation="LaTeX">\mathcal {AB}m</tex-math></inline-formula> has a global <inline-formula><tex-math notation="LaTeX">R</tex-math></inline-formula>-linear rate when the largest step-size and momentum parameter are positive and sufficiently small. We numerically show that <inline-formula><tex-math notation="LaTeX">\mathcal {AB}m</tex-math></inline-formula> achieves acceleration, particularly when the objective functions are ill-conditioned.