ARock: an Algorithmic Framework for Asynchronous Parallel Coordinate Updates (1506.02396v5)

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 At each step of ARock, an agent updates a randomly selected coordinate $x_i$ based on possibly out-of-date information on $x$.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.

• The paper introduces an asynchronous framework that updates coordinates in parallel, reducing synchronization delays while ensuring convergence with probability one.
• It employs fixed-point theory and stochastic analysis to guarantee convergence even in non-blocking, distributed computational settings.
• The framework is applied to various problems, from linear systems to convex optimization, demonstrating nearly linear speedup in practical experiments.

Asynchronous Parallel Coordinate Updates: An Analytical Overview

The paper, titled "ARock: an Algorithmic Framework for Asynchronous Parallel Coordinate Updates," presents a theoretical and practical analysis of using an asynchronous framework called ARock for parallel coordinate updates. This work encompasses advances in algorithmic structures for computational tasks, emphasizing a non-blocking approach to parallel computing, which promises reduced synchronization delays and enhanced computational speed.

Algorithmic Foundations and Methodology

At its core, ARock aims to find fixed points of nonexpansive operators, which are mathematical constructs central to numerous problems across numerical linear algebra, optimization, and scientific computations. The proposed framework operates by allowing multiple computational agents such as machines or processor cores to update coordinates in a parallel manner, inherently leveraging asynchrony to bypass synchronization bottlenecks.

The algorithm proceeds through a sequence of updates wherein each agent, at any given step, selects a coordinate randomly to update based on possibly outdated information and updates the shared variable in a lock-free environment when atomic operations are possible. Crucially, these updates do not require a global consensus on the state of computation, allowing simultaneous operations that sidestep traditional parallel processing hurdles.

Convergence and Theoretical Insights

The paper establishes robust theoretical foundations through convergence proofs for the ARock framework. The authors demonstrate that under weak conditions of the operator being nonexpansive and possessing a fixed point, the ARock sequence converges to a solution with probability one. The work also illuminates the framework's adaptability to various computational problems, achieving linear convergence under stronger assumptions, such as when dealing with quasi-strongly monotone operators.

To support these claims, detailed analytic techniques involving fixed-point iteration, operator theory, and stochastic Fejér monotonicity are employed. These approaches enable convergence guarantees even in stochastic settings, making the framework suited to large, complex systems where distribution of data or tasks is inherent.

Practical Applications and Implications

The paper extends ARock's utility to several special cases, including but not limited to:

• Linear systems of equations.
• Convex and smooth optimization problems.
• Decentralized and distributed consensus optimization.

These applications demonstrate ARock's versatility across domains that demand efficient parallel processing. For instance, numerical experiments, such as on logistic regression tasks, illustrate almost-linear speedups in running time compared to traditional synchronous methods. Such empirical validations underscore ARock’s potential in practical scenarios where computational resources are abundant, but synchronization challenges persist.

Future Directions in Asynchronous Computing

This paper opens pathways for further exploration into asynchronous algorithms. Potential future directions could involve enhancing the model to incorporate more complex dependencies in updates or extending it to non-convex optimization problems. Additionally, understanding the influence of network architectures and communication overhead on the algorithm's performance can guide refinements to better adapt ARock to current and emerging computation platforms.

In conclusion, by covering both solid theoretical ground and practical efficiency, ARock represents a significant contribution to computational mathematics, particularly in how parallelism is executed at scale. The framework has the promise to accelerate a wide range of applications, providing a compelling direction for future research in asynchronous computing paradigms.