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We consider the fundamental problem of solving a large-scale system of linearequations. In particular, we consider the setting where a taskmaster intends tosolve the system in a distributed/federated fashion with the help of a set ofmachines, who each have a subset of the equations. Although there exist severalapproaches for solving this problem, missing is a rigorous comparison betweenthe convergence rates of the projection-based methods and those of theoptimization-based ones. In this paper, we analyze and compare these twoclasses of algorithms with a particular focus on the most efficient method fromeach class, namely, the recently proposed Accelerated Projection-BasedConsensus (APC) and the Distributed Heavy-Ball Method (D-HBM). To this end, wefirst propose a geometric notion of data heterogeneity called angularheterogeneity and discuss its generality. Using this notion, we bound andcompare the convergence rates of the studied algorithms and capture the effectsof both cross-machine and local data heterogeneity on these quantities. Ouranalysis results in a number of novel insights besides showing that APC is themost efficient method in realistic scenarios where there is a large dataheterogeneity. Our numerical analyses validate our theoretical results.

On the Effects of Data Heterogeneity on the Convergence Rates of Distributed Linear System Solvers