Exact Diffusion for Distributed Optimization and Learning-Part I: Algorithm Development

• Published in: IEEE transactions on signal processing Vol. 67; no. 3; pp. 708 - 723
• Main Authors: Yuan, Kun, Ying, Bicheng, Zhao, Xiaochuan, Sayed, Ali H.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01-02-2019; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Aggregates; Algorithms; Computer simulation; consensus; Convergence; Convex functions; Diffusion; Distributed optimization; doubly stochastic matrix; exact convergence; left-stochastic matrix; locally balanced policy; Machine learning; Optimization; Signal processing algorithms; Stability analysis; Strategy; Symmetric matrices
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2018.2875898

This paper develops a distributed optimization strategy with guaranteed exact convergence for a broad class of left-stochastic combination policies. The resulting exact diffusion strategy is shown in Part II of this paper to have a wider stability range and superior convergence performance than the EXTRA strategy. The exact diffusion method is applicable to locally balanced left-stochastic combination matrices which, compared to the conventional doubly stochastic matrix, are more general and able to endow the algorithm with faster convergence rates, more flexible step-size choices, and improved privacy-preserving properties. The derivation of the exact diffusion strategy relies on reformulating the aggregate optimization problem as a penalized problem and resorting to a diagonally weighted incremental construction. Detailed stability and convergence analyses are pursued in Part II of this paper and are facilitated by examining the evolution of the error dynamics in a transformed domain. Numerical simulations illustrate the theoretical conclusions.