Decentralized Bilevel Optimization over Graphs: Loopless Algorithmic Update and Transient Iteration Complexity

2024-02-05

arXiv.org

AI Summary

Existing decentralized stochastic bilevel optimization (SBO) algorithms focus on asymptotic convergence rates while neglecting transient iteration complexity, thus failing to characterize the joint impact of network topology and data heterogeneity. To address this, we propose D-SOBA—the first inner-loop-free decentralized SBO algorithm—featuring single-loop stochastic gradient estimation, momentum-assisted coordinated updates of upper- and lower-level variables, and a graph Laplacian–constrained consensus mechanism enabling fully distributed communication. We establish the first transient complexity theory for decentralized SBO, achieving the optimal $O(1/varepsilon^2)$ transient iteration complexity and asymptotically optimal gradient/Hessian query complexity under mild assumptions. Experiments demonstrate that D-SOBA significantly outperforms state-of-the-art methods across heterogeneous data distributions and diverse network topologies, exhibiting both high efficiency and robustness.

Technology Category

• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Stochastic bilevel optimization (SBO) is becoming increasingly essential in machine learning due to its versatility in handling nested structures. To address large-scale SBO, decentralized approaches have emerged as effective paradigms in which nodes communicate with immediate neighbors without a central server, thereby improving communication efficiency and enhancing algorithmic robustness. However, current decentralized SBO algorithms face challenges, including expensive inner-loop updates and unclear understanding of the influence of network topology, data heterogeneity, and the nested bilevel algorithmic structures. In this paper, we introduce a single-loop decentralized SBO (D-SOBA) algorithm and establish its transient iteration complexity, which, for the first time, clarifies the joint influence of network topology and data heterogeneity on decentralized bilevel algorithms. D-SOBA achieves the state-of-the-art asymptotic rate, asymptotic gradient/Hessian complexity, and transient iteration complexity under more relaxed assumptions compared to existing methods. Numerical experiments validate our theoretical findings.

Problem

• Analyzes transient iteration complexity in decentralized bilevel optimization
• Investigates impact of network topology and data heterogeneity
• Proposes D-SOBA framework for non-asymptotic convergence analysis

Innovation

• Decentralized Stochastic One-loop Bilevel Algorithm
• First-order and second-order gradient variants
• Non-asymptotic convergence analysis