On the mean square error of randomized averaging algorithms

TL;DR

This paper analyzes the mean square error in randomized averaging algorithms, providing bounds and showing that the deviation diminishes as network size increases, independent of convergence properties.

Contribution

It introduces a novel approach to bound the mean square deviation in randomized averaging algorithms, applicable even with weak or sparse interactions.

Findings

Deviation tends to zero as network size grows

Decay rate of deviation is at least inverse of network size

Applicable to algorithms with weakly correlated interactions

Abstract

This paper regards randomized discrete-time consensus systems that preserve the average "on average". As a main result, we provide an upper bound on the mean square deviation of the consensus value from the initial average. Then, we apply our result to systems where few or weakly correlated interactions take place: these assumptions cover several algorithms proposed in the literature. For such systems we show that, when the network size grows, the deviation tends to zero, and the speed of this decay is not slower than the inverse of the size. Our results are based on a new approach, which is unrelated to the convergence properties of the system.