# Tight bounds on the convergence rate of generalized ratio consensus algorithms

## TL;DR

This paper establishes tight bounds on how quickly generalized ratio consensus algorithms converge almost surely, linking convergence rates to spectral gaps of the underlying random matrix sequences.

## Contribution

It provides the first sharp upper bounds for the exponential convergence rate of generalized ratio consensus algorithms based on spectral gap analysis.

## Key findings

- Upper bounds for convergence rate are sharp.
- Convergence rate is characterized by spectral gap.
- Results extend and complement previous work.

## Abstract

The problems discussed in this paper are motivated by general ratio consensus algorithms, introduced by Kempe, Dobra, and Gehrke (2003) in a simple form as the push-sum algorithm, later extended by B\'en\'ezit et al. (2010) under the name weighted gossip algorithm. We consider a communication protocol described by a strictly stationary, ergodic, sequentially primitive sequence of non-negative 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 the paper 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 (2013) and Iutzeler, Ciblat and Hachem (2013).