Polynomial Filtering for Fast Convergence in Distributed Consensus

TL;DR

This paper introduces polynomial filtering algorithms to significantly accelerate convergence in distributed consensus problems by shaping the network matrix spectrum, applicable to both static and dynamic networks.

Contribution

It proposes a novel polynomial filtering approach with a semi-definite programming method to optimize convergence speed in distributed consensus algorithms.

Findings

Accelerates convergence rate compared to traditional methods.

Effective for both static and dynamic network topologies.

Simulation results confirm improved performance.

Abstract

In the past few years, the problem of distributed consensus has received a lot of attention, particularly in the framework of ad hoc sensor networks. Most methods proposed in the literature address the consensus averaging problem by distributed linear iterative algorithms, with asymptotic convergence of the consensus solution. The convergence rate of such distributed algorithms typically depends on the network topology and the weights given to the edges between neighboring sensors, as described by the network matrix. In this paper, we propose to accelerate the convergence rate for given network matrices by the use of polynomial filtering algorithms. The main idea of the proposed methodology is to apply a polynomial filter on the network matrix that will shape its spectrum in order to increase the convergence rate. Such an algorithm is equivalent to periodic updates in each of the sensors by aggregating a few of its previous estimates. We formulate the computation of the coefficients of the optimal polynomial as a semi-definite program that can be efficiently and globally solved for both static and dynamic network topologies. We finally provide simulation results that demonstrate the effectiveness of the proposed solutions in accelerating the convergence of distributed consensus averaging problems.