  In the realm of big data and machine learning, data-parallel, distributed
stochastic algorithms have drawn significant attention in the present
days.~While the synchronous versions of these algorithms are well understood in
terms of their convergence, the convergence analyses of their asynchronous
counterparts are not widely studied. In this paper, we propose and analyze a
{\it distributed, asynchronous parallel} SGD in light of solving an arbitrary
consistent linear system by reformulating the system into a stochastic
optimization problem as studied by Richt\'{a}rik and Tak\'{a}\~{c} in [35]. We
compare the convergence rates of our asynchronous SGD algorithm with the
synchronous parallel algorithm proposed by Richt\'{a}rik and Tak\'{a}\v{c} in
[35] under different choices of the hyperparameters---the stepsize, the damping
factor, the number of processors, and the delay factor. We show that our
asynchronous parallel SGD algorithm also enjoys a global linear convergence
rate, similar to the {\em basic} method and the synchronous parallel method in
[35] for solving any arbitrary consistent linear system via stochastic
reformulation. We also show that our asynchronous parallel SGD improves upon
the {\em basic} method with a better convergence rate when the number of
processors is larger than four. We further show that this asynchronous approach
performs asymptotically better than its synchronous counterpart for certain
linear systems. Moreover, for certain linear systems, we compute the minimum
number of processors required for which our asynchronous parallel SGD is
better, and find that this number can be as low as two for some ill-conditioned
problems.
