Chebyshev Polynomials in Distributed Consensus Applications

TL;DR

This paper introduces a novel distributed consensus algorithm leveraging Chebyshev polynomials, significantly accelerating convergence by utilizing second-order difference equations, with proven theoretical and experimental benefits.

Contribution

The paper's main contribution is the development and analysis of a new consensus algorithm based on Chebyshev polynomials that improves convergence speed over existing methods.

Findings

Achieves faster consensus with fewer iterations.

Works effectively on fixed and switching topologies.

Validated through theoretical analysis and synthetic data experiments.

Abstract

In this paper we analyze the use of Chebyshev polynomials in distributed consensus applications. We study the properties of these polynomials to propose a distributed algorithm that reaches the consensus in a fast way. The algorithm is expressed in the form of a linear iteration and, at each step, the agents only require to transmit their current state to their neighbors. The difference with respect to previous approaches is that the update rule used by the network is based on the second order difference equation that describes the Chebyshev polynomials of first kind. As a consequence, we show that our algorithm achieves the consensus using far less iterations than other approaches. We characterize the main properties of the algorithm for both, fixed and switching communication topologies. The main contribution of the paper is the study of the properties of the Chebyshev polynomials in distributed consensus applications, proposing an algorithm that increases the convergence rate with respect to existing approaches. Theoretical results, as well as experiments with synthetic data, show the benefits using our algorithm.