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

TL;DR

This paper introduces Core-Halo decomposition for decentralized fixed-point problem solving, overcoming biases of strict decomposition by separating read and write operations, and achieves near-centralized performance.

Contribution

It proposes the Core-Halo decomposition method that faithfully implements fixed-point solutions in decentralized systems, addressing structural biases of strict decomposition.

Findings

Core-Halo achieves near-centralized performance in experiments.

Strict decomposition introduces irreducible structural bias.

Core-Halo effectively manages read-write dependencies in decentralized updates.

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.