The asymptotical error of broadcast gossip averaging algorithms

TL;DR

This paper analyzes the error behavior of broadcast gossip algorithms for distributed averaging, providing bounds and showing that errors diminish as network size increases, with implications for network estimation accuracy.

Contribution

It offers a theoretical framework using martingale theory to bound accumulated errors in broadcast gossip algorithms based on network properties.

Findings

Expected error can be bounded using network parameters.

Accumulated error tends to zero as network size grows.

Theoretical bounds are supported by simulations.

Abstract

In problems of estimation and control which involve a network, efficient distributed computation of averages is a key issue. This paper presents theoretical and simulation results about the accumulation of errors during the computation of averages by means of iterative "broadcast gossip" algorithms. Using martingale theory, we prove that the expectation of the accumulated error can be bounded from above by a quantity which only depends on the mixing parameter of the algorithm and on few properties of the network: its size, its maximum degree and its spectral gap. Both analytical results and computer simulations show that in several network topologies of applicative interest the accumulated error goes to zero as the size of the network grows large.