Coordinate Descent with Arbitrary Sampling I: Algorithms and Complexity
This work addresses the need for more flexible optimization algorithms in machine learning by introducing a method that generalizes existing coordinate descent variants, though it is incremental in extending sampling capabilities.
The paper tackles the problem of minimizing a sum of a smooth convex function and a convex block-separable regularizer by proposing ALPHA, a new randomized coordinate descent method that updates random subsets of coordinates based on an arbitrary distribution, with complexity bounds that match or improve upon the best known results.
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.