  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.

Machine Learning;Distributed, Parallel, and Cluster Computing;Numerical Analysis