Asynchronous Systems and Binary Diagonal Random Matrices: A Proof and Convergence Rate

In a synchronized network of $n$ nodes, each node will update its parameter based on the system state in a given iteration. It is well-known that the updates can converge to a fixed point if the maximum absolute eigenvalue (spectral radius) of the $n \times n$ iterative matrix ${\bf{F}}$ is less than one (i.e. $\rho({\bf{F}})<1$). However, if only a subset of the nodes update their parameter in an iteration (due to delays or stale feedback) then this effectively renders the spectral radius of the iterative matrix as one. We consider matrices of unit spectral radii generated from ${\bf{F}}$ due to random delays in the updates. We show that if each node updates at least once in every $T$ iterations, then the product of the random matrices (joint spectral radius) corresponding to these iterations is less than one. We then use this property to prove convergence of asynchronous iterative systems. Finally, we show that the convergence rate of such a system is $\rho({\bf{F}})\frac{(1-(1-\gamma)^T)^n}{T}$, where assuming ergodicity, $\gamma$ is the lowest bound on the probability that a node will update in any given iteration.