AsySPA: An Exact Asynchronous Algorithm for Convex Optimization Over Digraphs

TL;DR

This paper introduces AsySPA, an asynchronous distributed algorithm for convex optimization over directed graphs, capable of converging to the optimal solution despite asynchronous updates and communication delays.

Contribution

The paper presents the first exact asynchronous subgradient-push algorithm with adaptive stepsizes for directed graphs, ensuring convergence under arbitrary update rates and delays.

Findings

Proves asymptotic convergence to optimal solutions.

Provides explicit convergence rate analysis.

Demonstrates effectiveness through simulations.

Abstract

This paper proposes a novel exact distributed asynchronous subgradient-push algorithm (AsySPA) to solve an additive cost optimization problem over directed graphs where each node only has access to a local convex function and updates asynchronously with an arbitrary rate. Specifically, each node of a strongly connected digraph does not wait for updates from other nodes but simply starts a new update within any bounded time interval by using local information available from its in-neighbors. "Exact" means that every node of the AsySPA can asymptotically converge to the same optimal solution, even under different update rates among nodes and bounded communication delays. To address uneven update rates, we design a simple mechanism to adaptively adjust stepsizes per update in each node, which is substantially different from the existing works. Then, we construct a delay-free augmented system to address asynchrony and delays, and study its convergence by proposing a generalized subgradient algorithm, which clearly has its own significance and helps us to explicitly evaluate the convergence rate of the AsySPA. Finally, we demonstrate advantages of the AsySPA in both theory and simulation.