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

• URL: http://arxiv.org/abs/2304.10640v2
• Date: Fri, 16 Feb 2024 00:02:49 GMT
• Title: On the Effects of Data Heterogeneity on the Convergence Rates of Distributed Linear System Solvers
• Authors: Boris Velasevic, Rohit Parasnis, Christopher G. Brinton, Navid Azizan
• Abstract summary: We consider the problem of solving a large-scale system of linear equations with the help of a set of machines. We compare two classes of algorithms with a particular focus on the most efficient method from each class. Our analysis results in a number of novel insights besides showing that APC is the most efficient method in realistic scenarios.
• Score: 10.103350854870992
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider the fundamental problem of solving a large-scale system of linear equations. In particular, we consider the setting where a taskmaster intends to solve the system in a distributed/federated fashion with the help of a set of machines, who each have a subset of the equations. Although there exist several approaches for solving this problem, missing is a rigorous comparison between the convergence rates of the projection-based methods and those of the optimization-based ones. In this paper, we analyze and compare these two classes of algorithms with a particular focus on the most efficient method from each class, namely, the recently proposed Accelerated Projection-Based Consensus (APC) and the Distributed Heavy-Ball Method (D-HBM). To this end, we first propose a geometric notion of data heterogeneity called angular heterogeneity and discuss its generality. Using this notion, we bound and compare the convergence rates of the studied algorithms and capture the effects of both cross-machine and local data heterogeneity on these quantities. Our analysis results in a number of novel insights besides showing that APC is the most efficient method in realistic scenarios where there is a large data heterogeneity. Our numerical analyses validate our theoretical results.

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