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

TL;DR

This paper introduces an accelerated distributed consensus algorithm for solving large-scale linear systems, demonstrating faster convergence and speed-up over existing methods through theoretical analysis and experiments.

Contribution

It presents a novel accelerated consensus algorithm with projection-based updates and momentum, outperforming traditional distributed methods in convergence speed.

Findings

Faster convergence compared to gradient descent, Nesterov, and ADMM.

Significant speed-up on real-world data sets.

A new variation of distributed heavy-ball method with similar convergence rate.

Abstract

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