Tight Bounds on the Convergence Rate of Generalized Ratio Consensus Algorithms

The problems discussed in this paper are motivated by the ratio consensus problem formulated and solved in the context of the push-sum algorithm proposed in Kempe et al. (2003), and extended in Benezit et al. (2010) under the name weighted gossip algorithm. We consider a strictly stationary, ergodic, sequentially primitive sequence of $p \times p$ random matrices with non-negative entries $(A_n), ~n \ge 1.$ Let $x, w \in \mathbb R^p$ denote a pair of initial vectors, such that $w \ge 0, w \neq 0$. Our objective is to study the asymptotic properties of the ratios \begin{equation} e_i^\top A_n A_{n-1} \cdots A_1 x/ e_i^\top A_n A_{n-1} \cdots A_1 w, \qquad i=1,...,p, \end{equation} where $e_i$ is the unit vector with a single $1$ in its $i$-th coordinate. The main results of the paper provide upper bounds for the almost sure exponential convergence rate in terms of the spectral gap associated with $(A_n).$ It will be shown that these upper bounds are sharp. In the final section of the paper we present a variety of connections between the spectral gap of $(A_n)$ and the Birkhoff contraction coefficient of the product $A_n \cdots A_1.$ Our results complement previous results of Picci and Taylor (2013), and Tahbaz-Salehi and Jadbabaie (2010).