  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.
