Distributed Solution of Large-Scale Linear Systems via Accelerated Projection-Based Consensus

Solving a large-scale system of linear equations is a key step at the heart\nof many algorithms in machine learning, scientific computing, and beyond. When\nthe problem dimension is large, computational and/or memory constraints make it\ndesirable, or even necessary, to perform the task in a distributed fashion. In\nthis paper, we consider a common scenario in which a taskmaster intends to\nsolve a large-scale system of linear equations by distributing subsets of the\nequations among a number of computing machines/cores. We propose an accelerated\ndistributed consensus algorithm, in which at each iteration every machine\nupdates its solution by adding a scaled version of the projection of an error\nsignal onto the nullspace of its system of equations, and where the taskmaster\nconducts an averaging over the solutions with momentum. The convergence\nbehavior of the proposed algorithm is analyzed in detail and analytically shown\nto compare favorably with the convergence rate of alternative distributed\nmethods, namely distributed gradient descent, distributed versions of\nNesterov's accelerated gradient descent and heavy-ball method, the block\nCimmino method, and ADMM. On randomly chosen linear systems, as well as on\nreal-world data sets, the proposed method offers significant speed-up relative\nto all the aforementioned methods. Finally, our analysis suggests a novel\nvariation of the distributed heavy-ball method, which employs a particular\ndistributed preconditioning, and which achieves the same theoretical\nconvergence rate as the proposed consensus-based method.\n