Tight Bounds on the Convergence Rate of Generalized Ratio Consensus Algorithms

Balazs Gerencser,Laszlo Gerencser

doi:10.1109/tac.2021.3067629

Abstract

The problems discussed in this article are motivated by general ratio consensus algorithms, introduced by Kempe <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> in 2003 in a simple form as the push-sum algorithm, later extended by Bénézit <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> in 2013.

Highlights

The key results of this paper are stated as Theorems 12, 14, 16 and 19, extending previous results on the almost sure exponential convergence in the context of ratio consensus such as given in [9] and [10], in particular providing upper bounds for the almost sure exponential convergence rate in terms of spectral gaps associated with stationary sequences of matrices
In the last two sections of the main body of the paper we elaborate on the major mathematical details: in Section VII we describe the essential fabric of the proofs of the main theorems, while in Section VIII an interlude on the connection between spectral gap and Birkhoff’s contraction coefficient is added
First we describe the critical steps of the proofs of our main theorems, with some technical details relegated to Appendix IV, and a mathematical interlude on the spectral gap is added

Summary

Introduction

The problems discussed in this paper are motivated by the study of general ratio consensus algorithms, introduced in [1] in a simple form as the push-sum algorithm, and later extended in [2] under the name weighted gossip algorithm for solving a class of distributed computation problems. A nice application of the push-sum algorithm for computing the eigenvectors of a large symmetric matrix, corresponding to the adjacency matrix of an undirected graph, was given in [5]. Another application is distributed convex optimization, see [6].

Objectives

Findings

Discussion

Conclusion