ARock: an Algorithmic Framework for Asynchronous Parallel Coordinate Updates

TL;DR

ARock is an asynchronous parallel algorithmic framework for fixed-point problems that reduces synchronization bottlenecks, converges with weaker conditions, and applies to various scientific computing tasks including optimization and machine learning.

Contribution

We introduce ARock, a novel asynchronous parallel framework for fixed-point problems with proven convergence under weaker conditions and broad applicability.

Findings

ARock converges to fixed points with probability one.

Linear convergence is established under certain conditions.

Numerical experiments demonstrate effectiveness in sparse logistic regression.

Abstract

Finding a fixed point to a nonexpansive operator, i.e., $x^*=Tx^*$, abstracts many problems in numerical linear algebra, optimization, and other areas of scientific computing. To solve fixed-point problems, we propose ARock, an algorithmic framework in which multiple agents (machines, processors, or cores) update $x$ in an asynchronous parallel fashion. Asynchrony is crucial to parallel computing since it reduces synchronization wait, relaxes communication bottleneck, and thus speeds up computing significantly. At each step of ARock, an agent updates a randomly selected coordinate $x_i$ based on possibly out-of-date information on $x$. The agents share $x$ through either global memory or communication. If writing $x_i$ is atomic, the agents can read and write $x$ without memory locks. Theoretically, we show that if the nonexpansive operator $T$ has a fixed point, then with probability one, ARock generates a sequence that converges to a fixed points of $T$. Our conditions on $T$ and step sizes are weaker than comparable work. Linear convergence is also obtained. We propose special cases of ARock for linear systems, convex optimization, machine learning, as well as distributed and decentralized consensus problems. Numerical experiments of solving sparse logistic regression problems are presented.