Decentralized Composite Optimization in Stochastic Networks: A Dual Averaging Approach with Linear Convergence

TL;DR

This paper introduces a novel decentralized dual averaging algorithm capable of solving composite convex optimization problems over stochastic networks with proven global linear convergence under mild conditions, improving upon prior sublinear methods.

Contribution

The paper presents the first linearly convergent decentralized dual averaging algorithm for stochastic networks, utilizing a new dynamic consensus protocol for improved accuracy.

Findings

Achieves global linear convergence in stochastic networks.

Outperforms existing sublinear DDA algorithms.

Validated by numerical experiments.

Abstract

Decentralized optimization, particularly the class of decentralized composite convex optimization (DCCO) problems, has found many applications. Due to ubiquitous communication congestion and random dropouts in practice, it is highly desirable to design decentralized algorithms that can handle stochastic communication networks. However, most existing algorithms for DCCO only work in networks that are deterministically connected during bounded communication rounds, and therefore cannot be extended to stochastic networks. In this paper, we propose a new decentralized dual averaging (DDA) algorithm that can solve DCCO in stochastic networks. Under a rather mild condition on stochastic networks, we show that the proposed algorithm attains global linear convergence if each local objective function is strongly convex. Our algorithm substantially improves the existing DDA-type algorithms as the latter were only known to converge sublinearly prior to our work. The key to achieving the improved rate is the design of a novel dynamic averaging consensus protocol for DDA, which intuitively leads to more accurate local estimates of the global dual variable. To the best of our knowledge, this is the first linearly convergent DDA-type decentralized algorithm and also the first algorithm that attains global linear convergence for solving DCCO in stochastic networks. Numerical results are also presented to support our design and analysis.