Convergence Rate of Accelerated Average Consensus With Local Node Memory: Optimization and Analytic Solutions

Previous research works have shown that adding local memory can accelerate the consensus. It is natural to question what is the fastest rate achievable by the
$M$
-tap memory acceleration, and what are the corresponding control parameters. This article introduces a set of effective and previously unused techniques to analyze the convergence rate of accelerated consensus with
$M$
-tap memory of local nodes and to design the control protocols. These effective techniques, including the Kharitonov stability theorem, the Routh stability criterion, and the robust stability margin, have led to the following new results: first, the direct link between the convergence rate and the control parameters; second, explicit formulas of the optimal convergence rate and the corresponding optimal control parameters for
$M \leq 2$
on a given graph; third, analytic formulas of the optimal worst-case convergence rate and the corresponding optimal control parameters for the memory
$M \geq 1$
on a set of uncertain graphs. We show that the acceleration with the memory
$M=1$
provides the optimal convergence rate in the sense of the worst-case performance. Several numerical examples are given to demonstrate the validity and performance of the theoretical results.