  We develop and analyze an asynchronous algorithm for distributed convex
optimization when the objective writes a sum of smooth functions, local to each
worker, and a non-smooth function. Unlike many existing methods, our
distributed algorithm is adjustable to various levels of communication cost,
delays, machines computational power, and functions smoothness. A unique
feature is that the stepsizes do not depend on communication delays nor number
of machines, which is highly desirable for scalability. We prove that the
algorithm converges linearly in the strongly convex case, and provide
guarantees of convergence for the non-strongly convex case. The obtained rates
are the same as the vanilla proximal gradient algorithm over some introduced
epoch sequence that subsumes the delays of the system. We provide numerical
results on large-scale machine learning problems to demonstrate the merits of
the proposed method.
