Towards an $O(\frac{1}{t})$ convergence rate for distributed dual averaging

TL;DR

This paper introduces a second-order consensus scheme for distributed dual averaging that achieves an improved $O(1/t)$ convergence rate for smooth problems, outperforming previous methods with $O(1/\sqrt{t})$ rate.

Contribution

It proposes a novel second-order consensus scheme combined with inexact gradient oracles to attain faster convergence in distributed dual averaging for smooth problems.

Findings

Achieves $O(1/t)$ convergence rate for smooth distributed dual averaging.

Effectively bounds consensus error accumulation over time.

Demonstrates effectiveness in large-scale LASSO problem.

Abstract

Recently, distributed dual averaging has received increasing attention due to its superiority in handling constraints and dynamic networks in multiagent optimization. However, all distributed dual averaging methods reported so far considered nonsmooth problems and have a convergence rate of $O(\frac{1}{\sqrt{t}})$. To achieve an improved convergence guarantee for smooth problems, this work proposes a second-order consensus scheme that assists each agent to locally track the global dual variable more accurately. This new scheme in conjunction with smoothness of the objective ensures that the accumulation of consensus error over time caused by incomplete global information is bounded from above. Then, a rigorous investigation of dual averaging with inexact gradient oracles is carried out to compensate the consensus error and achieve an $O(\frac{1}{t})$ convergence rate. The proposed method is examined in a large-scale LASSO problem.