Push-sum Distributed Dual Averaging for Convex Optimization in Multi-agent Systems with Communication Delays

TL;DR

This paper introduces a push-sum distributed dual averaging algorithm for convex optimization in multi-agent systems with communication delays, proving its convergence and analyzing the impact of delays on performance.

Contribution

It proposes a novel algorithm that handles communication delays in distributed convex optimization and provides theoretical convergence guarantees.

Findings

Convergence rate of $ ext{O}(T^{-0.5})$ with proper step size.

Error bound depends on maximum communication delay.

Numerical simulations confirm theoretical results.

Abstract

The distributed convex optimization problem over the multi-agent system is considered in this paper, and it is assumed that each agent possesses its own cost function and communicates with its neighbours over a sequence of time-varying directed graphs. However, due to some reasons there exist communication delays while agents receive information from other agents, and we are going to seek the optimal value of the sum of agents' loss functions in this case. We desire to handle this problem with the push-sum distributed dual averaging (PS-DDA) algorithm. It is proved that this algorithm converges and the error decays at a rate $\mathcal{O}\left(T^{-0.5}\right)$ with proper step size, where $T$ is iteration span. The main result presented in this paper also illustrates the convergence of the proposed algorithm is related to the maximum value of the communication delay on one edge. We finally apply the theoretical results to numerical simulations to show the PS-DDA algorithm's performance.