Graph Balancing for Distributed Subgradient Methods over Directed Graphs

TL;DR

This paper introduces a distributed subgradient algorithm for multi-agent convex optimization over directed graphs, utilizing weight balancing to ensure convergence despite asymmetric communication, with proven convergence rates.

Contribution

It proposes a novel weight balancing technique for distributed subgradient methods over directed graphs, ensuring convergence and providing a new approach for average consensus.

Findings

Convergence rate of $O(rac{ ext{log} T}{ ext{sqrt} T})$ for objective and consensus.

Algorithm effectively handles directed communication asymmetries.

Special case yields a new distributed average consensus method.

Abstract

We consider a multi agent optimization problem where a set of agents collectively solves a global optimization problem with the objective function given by the sum of locally known convex functions. We focus on the case when information exchange among agents takes place over a directed network and propose a distributed subgradient algorithm in which each agent performs local processing based on information obtained from his incoming neighbors. Our algorithm uses weight balancing to overcome the asymmetries caused by the directed communication network, i.e., agents scale their outgoing information with dynamically updated weights that converge to balancing weights of the graph. We show that both the objective function values and the consensus violation, at the ergodic average of the estimates generated by the algorithm, converge with rate $O(\frac{\log T}{\sqrt{T}})$, where $T$ is the number of iterations. A special case of our algorithm provides a new distributed method to compute average consensus over directed graphs.