Tight Bounds on the Convergence Rate of Generalized Ratio Consensus Algorithms

• Published in: IEEE transactions on automatic control Vol. 67; no. 4; pp. 1669 - 1684
• Main Authors: Gerencser, Balazs, Gerencser, Laszlo
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.04.2022; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Asymptotic properties; Asynchronous communication; communication networks; Consensus algorithm; Convergence; distributed computing; estimation; Heuristic algorithms; iterative algorithms; Mathematical analysis; multiagent systems; Nodes; Protocols; random processes; spectral gap; Symmetric matrices; Technological innovation; Upper bound; Upper bounds
• ISSN: 0018-9286; 1558-2523; 2334-3303; 1558-2523
• DOI: 10.1109/TAC.2021.3067629

The problems discussed in this article are motivated by general ratio consensus algorithms, introduced by Kempe et al. in 2003 in a simple form as the push-sum algorithm, later extended by Bénézit et al. in 2010 under the name weighted gossip algorithm. We consider a communication protocol described by a strictly stationary, ergodic, sequentially primitive sequence of nonnegative matrices, applied iteratively to a pair of fixed initial vectors, the components of which are called values and weights defined at the nodes of a network. The subject of ratio consensus problems is to study the asymptotic properties of ratios of values and weights at each node, expecting convergence to the same limit for all nodes. The main results of this article provide upper bounds for the rate of the almost sure exponential convergence in terms of the spectral gap associated with the given sequence of random matrices. It will be shown that these upper bounds are sharp. Our results complement previous results of Picci and Taylor in 2013 and Iutzeler et al. in 2013.