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

Abstract

Previous researches have shown that adding local memory can accelerate the consensus. It is natural to question what is the fastest rate achievable by the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M$</tex-math></inline-formula> -tap memory acceleration, and what are the corresponding control parameters. This paper introduces a set of effective and previously unused techniques to analyze the convergence rate of accelerated consensus with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M$</tex-math></inline-formula> -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: 1) the direct link between the convergence rate and the control parameters; 2) explicit formulas of the optimal convergence rate and the corresponding optimal control parameters for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M \leq 2$</tex-math></inline-formula> on a given graph; 3) analytic formulas of the optimal worst-case convergence rate and the corresponding optimal control parameters for the memory <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M \geq 1$</tex-math></inline-formula> on a set of uncertain graphs. We show that the acceleration with the memory <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M=1$</tex-math></inline-formula> provides the optimal convergence rate in the sense of worst-case performance. Several numerical examples are given to demonstrate the validity and performance of the theoretical results.