Push-Pull Gradient Methods for Distributed Optimization in Networks (1810.06653v4)

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 new distributed gradient-based methods 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 gradients is pushed to the neighbors, while the information about the decision variable is pulled from the neighbors hence giving the name "push-pull gradient methods". The methods utilize two different graphs for the information exchange among agents, and as such, unify 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 proposed algorithms and their many variants converge linearly for strongly convex and smooth objective functions over a network (possibly with unidirectional data links) in both synchronous and asynchronous random-gossip settings. In particular, under the random-gossip setting, "push-pull" is the first class of algorithms for distributed optimization over directed graphs. Moreover, we numerically evaluate our proposed algorithms in both scenarios, and show that they outperform other existing linearly convergent schemes, especially for ill-conditioned problems and networks that are not well balanced.

• The paper introduces push-pull gradient methods that decentralize information exchange, ensuring linear convergence for distributed convex problems.
• The methodology leverages a 'push' of gradient data and a 'pull' of decision variables, unifying performance across various network architectures.
• Numerical evaluations demonstrate superior performance in asynchronous and directed networks, highlighting efficacy in handling poorly conditioned and imbalanced systems.

Push-Pull Gradient Methods for Distributed Optimization in Networks

The paper under consideration presents a novel approach to solving distributed convex optimization problems in networked systems. Each agent in the network has a unique convex cost function, and the collective goal is to minimize the sum of these functions while adhering to the constraints imposed by the network's connectivity structure. This research introduces distributed gradient-based methods dubbed "Push-Pull Gradient Methods," characterized by each node maintaining two estimates: one for the decision variable and another for the gradient of the average objective function.

Methodology

The paper focuses on a key distinction in information dissemination: gradient information is "pushed" to neighboring nodes, whereas decision variable information is "pulled" from neighbors. This dichotomy is reflected in the name "Push-Pull Gradient Methods." Two separate graphs facilitate the exchange of information between agents, allowing the algorithm to adapt to various distributed architectures, such as fully decentralized, centralized, and semi-centralized systems. This flexibility helps unify approaches traditionally tailored to different network structures.

A significant contribution of this work is the proposition that the proposed algorithms converge linearly for strongly convex and smooth objective functions even in networks with unidirectional data links. This is shown to hold true under both synchronous and asynchronous random-gossip settings. In particular, the random-gossip "push-pull" algorithm emerges as a novel class within directed graph-based distributed optimization algorithms. The paper underlines this with numerical evaluations demonstrating superior performance over existing linearly convergent methods, particularly in addressing poorly conditioned problems and imbalanced networks.

Implications and Future Directions

From a practical standpoint, this research offers a robust framework for distributed optimization that is adaptable to a diverse array of network architectures. The algorithms' linear convergence further enhances their applicability in time-sensitive tasks. Theoretically, this work paves the way for further exploration into distributed optimization methods that can handle greater variability in network topology and dynamics.

Future research may explore refining these methods to handle non-convex objectives or more complex network conditions. Additionally, a comprehensive understanding of the underlying graph's role in the optimization process could lead to more effective algorithm designs that are inherently resilient to network changes or disruptions.

Conclusion

Overall, this paper contributes significantly to the field of distributed optimization through its introduction of push-pull gradient methods. Its formulation and analysis establish the foundation for algorithms that competently navigate the challenges posed by distributed, networked systems. This work not only advances the theoretical understanding of distributed optimization but also offers practical solutions that can be deployed in real-world, network-based applications.