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

arXiv: https://arxiv.org/abs/2605.08681 · HTML: https://mlsanity.com/paper/2605.08681/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 55 references that are in the map.
- First posted on arXiv 9 May 2026.

## Builds on (most cited first, 19 of 19)

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- [PageRank beyond the Web](https://mlsanity.com/paper/1407.5107/) (2014)
- [Decentralized Multi-Agent Reinforcement Learning with Networked Agents: Recent Advances](https://mlsanity.com/paper/1912.03821/) (2019)
- [Decentralized Reinforcement Learning: Global Decision-Making via Local Economic Transactions](https://mlsanity.com/paper/2007.02382/) (2020)
- [Finite-Time Convergence Rates of Decentralized Stochastic Approximation with Applications in Multi-Agent and Multi-Task Learning](https://mlsanity.com/paper/2010.15088/) (2020)
- [Understanding Representation of Deep Equilibrium Models from Neural Collapse Perspective](https://mlsanity.com/paper/2410.23391/) (2024)
- [Distributed Stochastic Approximation: Weak Convergence and Network Design](https://mlsanity.com/paper/1412.3158/) (2014)

## Abstract (arXiv)

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.