Centroid Approximation with Multidimensional Approximate Agreement Protocols

Abstract

In this paper, we present distributed fault-tolerant algorithms that approximate the centroid (i.e., the average) of a set of $n$ data points in $\mathbb{R}^d$. Our work falls into the broader area of multidimensional Byzantine approximate agreement. We show that state-of-the-art algorithms, such as agreeing inside the convex hull of all non-faulty vectors, or minimum-diameter averaging (MDA), in the worst case either prevent us from agreeing on a vector close to the centroid (in terms of approximation quality), or allow Byzantine parties to influence the output considerably (in terms of validity). To design better approximation algorithms, we propose a novel concept of defining an approximation ratio of the centroid by including the vectors of the Byzantine adversaries in the definition. We analyze the algorithms in the synchronous and asynchronous models of communication with public communication channels. We show that the standard agreement algorithms based on agreeing inside the convex hull of all non-faulty vectors do not allow us to compute a better approximation than $2d$ of the centroid. On the other hand, MDA can be used to achieve constant approximation at the cost of only satisfying strong validity. As a trade-off, we develop an approach that reaches a $2\sqrt{d}$-approximation of the centroid, while satisfying box validity. Our approach provides optimal resilience, allowing up to $t<n/3$ faulty nodes.