This paper studies decentralized bilevel optimization, in which multiple agents collaborate to solve problems involving nested optimization structures with neighborhood communications. Most existing literature primarily utilizes gradient tracking to mitigate the influence of data heterogeneity, without exploring other well-known heterogeneity-correction techniques such as EXTRA or Exact Diffusion. Additionally, these studies often employ identical decentralized strategies for both upper- and lower-level problems, neglecting to leverage distinct mechanisms across different levels. To address these limitations, this paper proposes SPARKLE, a unified Single-loop Primal-dual AlgoRithm frameworK for decentraLized bilEvel optimization. SPARKLE offers the flexibility to incorporate various heterogeneitycorrection strategies into the algorithm. Moreover, SPARKLE allows for different strategies to solve upper- and lower-level problems. We present a unified convergence analysis for SPARKLE, applicable to all its variants, with state-of-the-art convergence rates compared to existing decentralized bilevel algorithms. Our results further reveal that EXTRA and Exact Diffusion are more suitable for decentralized bilevel optimization, and using mixed strategies in bilevel algorithms brings more benefits than relying solely on gradient tracking.

翻译：本文研究去中心化双层优化问题，其中多个智能体通过邻域通信协作解决具有嵌套优化结构的问题。现有文献主要利用梯度跟踪来缓解数据异质性的影响，而未探索其他已知的异质性校正技术（如EXTRA或精确扩散法）。此外，这些研究通常对上层和下层问题采用相同的去中心化策略，未能利用不同层级间的差异化机制。为克服这些局限，本文提出SPARKLE——一种用于去中心化双层优化的统一单循环原始对偶算法框架。该框架具备灵活性，可将多种异质性校正策略融入算法设计。同时，SPARKLE允许针对上下层问题采用不同求解策略。我们提出了适用于所有变体的统一收敛性分析，其收敛速率相较于现有去中心化双层算法达到最优水平。研究结果进一步表明：EXTRA与精确扩散法更适用于去中心化双层优化场景，且在双层算法中采用混合策略比单纯依赖梯度跟踪能带来更多优势。