Distributed SDDM Solvers: Theory & Applications

TL;DR

This paper introduces distributed solvers for symmetric diagonally dominant M-matrices, enabling faster and more accurate network flow optimization with theoretical convergence guarantees and superior empirical performance.

Contribution

It presents novel distributed SDDM solvers with different communication constraints and applies them to develop an efficient Newton method for network flow optimization.

Findings

Our solvers outperform existing methods in convergence speed.

The Newton method achieves superlinear convergence near the optimum.

Experimental results show significant performance improvements on test networks.

Abstract

In this paper, we propose distributed solvers for systems of linear equations given by symmetric diagonally dominant M-matrices based on the parallel solver of Spielman and Peng. We propose two versions of the solvers, where in the first, full communication in the network is required, while in the second communication is restricted to the R-Hop neighborhood between nodes for some $R \geq 1$. We rigorously analyze the convergence and convergence rates of our solvers, showing that our methods are capable of outperforming state-of-the-art techniques. Having developed such solvers, we then contribute by proposing an accurate distributed Newton method for network flow optimization. Exploiting the sparsity pattern of the dual Hessian, we propose a Newton method for network flow optimization that is both faster and more accurate than state-of-the-art techniques. Our method utilizes the distributed SDDM solvers for determining the Newton direction up to any arbitrary precision $\epsilon >0$. We analyze the properties of our algorithm and show superlinear convergence within a neighborhood of the optimal. Finally, in a set of experiments conducted on randomly generated and barbell networks, we demonstrate that our approach is capable of significantly outperforming state-of-the-art techniques.