  The need for scalable numerical solutions has motivated the development of
asynchronous parallel algorithms, where a set of nodes run in parallel with
little or no synchronization, thus computing with delayed information. This
paper studies the convergence of the asynchronous parallel algorithm ARock
under potentially unbounded delays.
  ARock is a general asynchronous algorithm that has many applications. It
parallelizes fixed-point iterations by letting a set of nodes randomly choose
solution coordinates and update them in an asynchronous parallel fashion. ARock
takes some recent asynchronous coordinate descent algorithms as special cases
and gives rise to new asynchronous operator-splitting algorithms. Existing
analysis of ARock assumes the delays to be bounded, and uses this bound to set
a step size that is important to both convergence and efficiency. Other work,
though allowing unbounded delays, imposes strict conditions on the underlying
fixed-point operator, resulting in limited applications.
  In this paper, convergence is established under unbounded delays, which can
be either stochastic or deterministic. The proposed step sizes are more
practical and generally larger than those in the existing work. The step size
adapts to the delay distribution or the current delay being experienced in the
system. New Lyapunov functions, which are the key to analyzing asynchronous
algorithms, are generated to obtain our results. A set of applicable
optimization algorithms with large-scale applications are given, including
machine learning and scientific computing algorithms.
