Core-Halo Decomposition: Decentralizing Large-Scale Fixed-Point Problems

2026-05-09

AI Summary

This work addresses the structural bias arising from strict block-wise decomposition when solving large-scale fixed-point equations in decentralized multi-agent systems. To mitigate this issue, the authors propose the Core-Halo decomposition method, which decouples write ownership from read context: each agent updates its core variables while reading overlapping halo variables. This design preserves parallelism while faithfully reconstructing the original problem. The approach innovatively introduces a Core-Halo structure to align operator dependencies, revealing the fundamental limitation of strict partitioning and establishing a lower bound on block-induced bias via a Bellman closure condition. Empirical results demonstrate that the framework achieves accuracy comparable to centralized solutions across diverse scenarios, while retaining the scalability and parallelism inherent to decentralized systems.

Abstract

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.

Problem

• fixed-point problems
• decentralized optimization
• structural bias
• block decomposition
• multi-agent systems

Innovation

• Core-Halo decomposition
• fixed-point problems
• decentralized optimization
• block-dependence structure
• structural bias