Asynchronous Distributed ADMM for Large-Scale Optimization- Part II: Linear Convergence Analysis and Numerical Performance

TL;DR

This paper analyzes the linear convergence of an asynchronous distributed ADMM algorithm for large-scale optimization, highlighting how network factors influence performance and demonstrating its efficiency on logistic regression tasks.

Contribution

It provides the first characterization of linear convergence conditions for asynchronous distributed ADMM, including the effects of network delay and size.

Findings

AD-ADMM achieves linear convergence under specific conditions.

The algorithm's performance is affected by network delay and size.

Numerical tests show superior efficiency on large-scale logistic regression.

Abstract

The alternating direction method of multipliers (ADMM) has been recognized as a versatile approach for solving modern large-scale machine learning and signal processing problems efficiently. When the data size and/or the problem dimension is large, a distributed version of ADMM can be used, which is capable of distributing the computation load and the data set to a network of computing nodes. Unfortunately, a direct synchronous implementation of such algorithm does not scale well with the problem size, as the algorithm speed is limited by the slowest computing nodes. To address this issue, in a companion paper, we have proposed an asynchronous distributed ADMM (AD-ADMM) and studied its worst-case convergence conditions. In this paper, we further the study by characterizing the conditions under which the AD-ADMM achieves linear convergence. Our conditions as well as the resulting linear rates reveal the impact that various algorithm parameters, network delay and network size have on the algorithm performance. To demonstrate the superior time efficiency of the proposed AD-ADMM, we test the AD-ADMM on a high-performance computer cluster by solving a large-scale logistic regression problem.