  Aiming at solving large-scale learning problems, this paper studies
distributed optimization methods based on the alternating direction method of
multipliers (ADMM). By formulating the learning problem as a consensus problem,
the ADMM can be used to solve the consensus problem in a fully parallel fashion
over a computer network with a star topology. However, traditional synchronized
computation does not scale well with the problem size, as the speed of the
algorithm is limited by the slowest workers. This is particularly true in a
heterogeneous network where the computing nodes experience different
computation and communication delays. In this paper, we propose an asynchronous
distributed ADMM (AD-AMM) which can effectively improve the time efficiency of
distributed optimization. Our main interest lies in analyzing the convergence
conditions of the AD-ADMM, under the popular partially asynchronous model,
which is defined based on a maximum tolerable delay of the network.
Specifically, by considering general and possibly non-convex cost functions, we
show that the AD-ADMM is guaranteed to converge to the set of
Karush-Kuhn-Tucker (KKT) points as long as the algorithm parameters are chosen
appropriately according to the network delay. We further illustrate that the
asynchrony of the ADMM has to be handled with care, as slightly modifying the
implementation of the AD-ADMM can jeopardize the algorithm convergence, even
under a standard convex setting.
