  We study the problem of minimizing the sum of a smooth convex function and a
convex block-separable regularizer and propose a new randomized coordinate
descent method, which we call ALPHA. Our method at every iteration updates a
random subset of coordinates, following an arbitrary distribution. No
coordinate descent methods capable to handle an arbitrary sampling have been
studied in the literature before for this problem. ALPHA is a remarkably
flexible algorithm: in special cases, it reduces to deterministic and
randomized methods such as gradient descent, coordinate descent, parallel
coordinate descent and distributed coordinate descent -- both in nonaccelerated
and accelerated variants. The variants with arbitrary (or importance) sampling
are new. We provide a complexity analysis of ALPHA, from which we deduce as a
direct corollary complexity bounds for its many variants, all matching or
improving best known bounds.
