[Paper Review] Geometric Convergence for Distributed Optimization with Barzilai-Borwein Step Sizes

Juan Gao, Xinwei Liu+3 more|arXiv (Cornell University)|Jul 18, 2019

Distributed Control Multi-Agent Systems51 references4 citations

TL;DR

This paper proposes DGM-BB-C, a distributed gradient method that uses Barzilai-Borwein (BB) step sizes to achieve geometric convergence to the exact optimal solution in multi-agent optimization. By combining an adapt-then-combine dynamic average consensus with multi-consensus inner loops, the method enables each agent to compute its own step size using only local information, allowing larger, adaptive step sizes while maintaining linear convergence without requiring increasing consensus steps.

ABSTRACT

We consider a distributed multi-agent optimization problem over a time-invariant undirected graph, where each agent possesses a local objective function and all agents collaboratively minimize the average of all objective functions through local computations and communications among neighbors. Recently, a class of distributed gradient methods has been proposed that achieves both exact and geometric convergence when a constant step size is used. The geometric convergence of these methods is ensured for conservatively selected step sizes, but how to choose an appropriate step size while running the algorithms has not been fully addressed. The Barzilai-Borwein (BB) method is a simple and effective technique for step sizes and requires few storage and inexpensive computations. It has been widely applied in various areas. In this paper, we introduce the BB method to distributed optimization. Based on an adapt-then-combine variation of the dynamic average consensus approach and using multi-consensus inner loops, we propose a distributed gradient method with BB step sizes (DGM-BB-C). Our method computes the step size for each agent automatically which only depends on its local information and is independent of that for other agents, and the larger step sizes are always permissible. Our method can seek the exact optimum when the number of consensus steps stays constant. We prove that DGM-BB-C has geometric convergence to the optimal solution. Simulation results on a distributed sensing problem show that our method is superior to some advanced methods in terms of iterations, gradient evaluations, communications and the related cost framework. These results validate our theoretical discoveries.

Motivation & Objective

• To address the challenge of selecting appropriate constant step sizes in distributed gradient methods that ensure exact and geometric convergence.
• To develop a distributed optimization algorithm that automatically computes step sizes per agent using only local information, avoiding global coordination.
• To enable larger, adaptive step sizes while maintaining linear convergence, overcoming limitations of fixed or diminishing step sizes.
• To achieve exact convergence with a constant number of consensus iterations, unlike methods requiring increasing inner loops.
• To validate the theoretical convergence with numerical experiments on distributed sensing and least squares problems.

Proposed method

• The method employs an adapt-then-combine variant of dynamic average consensus to track the average gradient across agents.
• It uses multi-consensus inner loops to improve gradient tracking accuracy and convergence stability.
• Each agent computes its own Barzilai-Borwein step size based on local gradient differences, eliminating dependence on other agents' step sizes.
• The BB step size rule is applied locally, allowing larger and adaptive step sizes that improve convergence speed.
• The algorithm is designed to maintain geometric convergence even with uncoordinated, larger step sizes.
• The method is proven to converge linearly to the exact solution under strong convexity and smoothness assumptions.

Experimental results

Research questions

• RQ1 Can Barzilai-Borwein step sizes be effectively adapted to distributed optimization to achieve geometric convergence with larger, adaptive step sizes?
• RQ2 Does the proposed method maintain exact convergence when the number of consensus steps is kept constant, unlike prior methods requiring increasing inner loops?
• RQ3 How does the performance of the BB-based distributed method compare to state-of-the-art algorithms in terms of iterations, gradient evaluations, and communication cost?
• RQ4 Can the convergence rate be theoretically bounded and empirically validated under varying network connectivity?
• RQ5 What is the relationship between the maximum BB step size and the convergence rate in the distributed setting?

Key findings

• DGM-BB-C achieves geometric convergence to the exact optimal solution for smooth and strongly convex functions, even with a constant number of consensus steps.
• The algorithm allows larger step sizes than existing methods, such as EXTRA and ATC-DIGing, which require step sizes below 1/(2L) or 2μ/L².
• Numerical results show DGM-BB-C requires fewer iterations, gradient evaluations, and communications than DGD, NEAR-DGD+, EXTRA, DIGing, and ATC-DIGing.
• For a network with connectivity probability 0.1, DGM-BB-C with R=4 achieved the best performance, while for 0.3, R=3 was optimal, indicating adaptability to network structure.
• The computed spectral radius ρ(G^α) was approximately 0.871, consistent with the theoretical convergence rate estimate.
• The maximum BB step size α_max reached 1.836, well within the theoretical bounds, confirming the robustness and practicality of the adaptive step size rule.