On the Convergence of Asynchronous Parallel Iteration with Unbounded Delays

TL;DR

This paper analyzes the convergence of asynchronous parallel algorithms with unbounded delays using a probabilistic approach, providing a practical stepsize formula based on delay statistics, and demonstrating improved convergence over existing methods.

Contribution

It introduces a convergence analysis for async-parallel methods with unbounded delays, deriving a stepsize formula based on delay distribution, validated by empirical delay measurements.

Findings

Delays follow a Poisson distribution with parameter p.

The proposed stepsize improves convergence speed.

Existing maximum-delay stepsizes are overly conservative.

Abstract

Recent years have witnessed the surge of asynchronous parallel (async-parallel) iterative algorithms due to problems involving very large-scale data and a large number of decision variables. Because of asynchrony, the iterates are computed with outdated information, and the age of the outdated information, which we call delay, is the number of times it has been updated since its creation. Almost all recent works prove convergence under the assumption of a finite maximum delay and set their stepsize parameters accordingly. However, the maximum delay is practically unknown. This paper presents convergence analysis of an async-parallel method from a probabilistic viewpoint, and it allows for large unbounded delays. An explicit formula of stepsize that guarantees convergence is given depending on delays' statistics. With $p+1$ identical processors, we empirically measured that delays closely follow the Poisson distribution with parameter $p$, matching our theoretical model, and thus the stepsize can be set accordingly. Simulations on both convex and nonconvex optimization problems demonstrate the validness of our analysis and also show that the existing maximum-delay induced stepsize is too conservative, often slowing down the convergence of the algorithm.