A Push-Pull Gradient Method for Distributed Optimization in Networks

TL;DR

This paper introduces a novel push-pull gradient method for distributed convex optimization in networks, unifying various architectures and demonstrating linear convergence under certain conditions.

Contribution

The paper proposes a new distributed gradient method that combines push and pull strategies, applicable to multiple network architectures, with proven linear convergence for strongly convex functions.

Findings

Linear convergence on directed static networks

Effective performance on time-varying directed networks

Unifies decentralized, centralized, and semi-centralized architectures

Abstract

In this paper, we focus on solving a distributed convex optimization problem in a network, where each agent has its own convex cost function and the goal is to minimize the sum of the agents' cost functions while obeying the network connectivity structure. In order to minimize the sum of the cost functions, we consider a new distributed gradient-based method where each node maintains two estimates, namely, an estimate of the optimal decision variable and an estimate of the gradient for the average of the agents' objective functions. From the viewpoint of an agent, the information about the decision variable is pushed to the neighbors, while the information about the gradients is pulled from the neighbors (hence giving the name "push-pull gradient method"). The method unifies the algorithms with different types of distributed architecture, including decentralized (peer-to-peer), centralized (master-slave), and semi-centralized (leader-follower) architecture. We show that the algorithm converges linearly for strongly convex and smooth objective functions over a directed static network. In our numerical test, the algorithm performs well even for time-varying directed networks.