提出核心-光环分解方法，让分布式系统更准确求解大规模固定点问题。
Core-Halo Decomposition: Decentralizing Large-Scale Fixed-Point Problems
- 将变量分为核心与重叠光环区，更新时只改核心，读取时包含周边信息。
- 实验证明该方法接近中心化性能，且保持分布式并行优势。
- 适用于需要高精度的分布式优化场景，如强化学习与大规模系统建模。
我们研究在分解框架下求解大规模固定点方程 $x^ar = \bar F(x^ar)$。标准严格分解将每个智能体分配互不重叠的变量块，并仅使用本块变量进行更新。然而，大多数算子的块更新依赖于块外变量。通过截断这些依赖关系，严格分解会改变平均算子，引入无法通过增加样本、减小步长或额外一致性机制消除的结构偏差。为此，我们提出核心-光环分解：将写入所有权与读取上下文分离，每个智能体更新自身核心，同时读取重叠的光环区域。通过将核心-光环结构与 $\bar F$ 的块依赖结构对齐，可在去中心化多智能体系统中忠实实现原固定点问题。我们进一步通过贝尔曼闭包条件和块级偏差下界刻画了严格分解的根本障碍，证明局部更新可改变原始固定点算子。最后，在多种应用设置下进行大量实验，结果表明核心-光环方法在保持去中心化并行性的同时，达到近似中心化的性能。
原文摘要 · Abstract (English)
We study solving large-scale fixed-point equation \(x^\star=\bar F(x^\star)\) with decomposition. Standard strict decomposition assigns each agent a disjoint block and evaluates updates using only owned coordinates. For most operators, however, a block update may depend on variables outside the block. Truncating these dependencies by strict decomposition changes the mean operator and creates structural bias that cannot be removed by more samples, smaller stepsizes, or additional consensus. We therefore propose Core-Halo decomposition, which separates write ownership from read-only evaluation context: each agent updates its own core and reads from an overlapping halo. By aligning the Core-Halo decomposition with the block-dependence structure of $\bar F$, the original fixed-point problem can be implemented faithfully in a decentralized multi-agent system. We further characterize the fundamental obstruction faced by strict decomposition through a Bellman closure condition and a blockwise bias lower bound, showing that local-only updates can alter the original fixed-point operator. Finally, we conduct extensive experiments across a range of application settings, and demonstrate that Core-Halo achieves near-centralized performance while retaining the parallelism benefits of decentralization.
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