Coordinate Descent with Arbitrary Sampling I: Algorithms and Complexity

TL;DR

This paper introduces ALPHA, a flexible randomized coordinate descent algorithm capable of handling arbitrary sampling distributions, unifying and improving upon existing methods for convex optimization with block-separable regularizers.

Contribution

The paper presents ALPHA, a novel coordinate descent method that supports arbitrary sampling, unifies various existing algorithms, and provides improved complexity bounds.

Findings

ALPHA generalizes multiple coordinate descent algorithms.

It achieves complexity bounds matching or better than existing methods.

Variants with importance sampling are introduced and analyzed.

Abstract

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.