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This arXiv paper gives theoretical conditions under which decentralized learners with different data distributions converge to shared distributions within groups defined by symmetries of the data and communication graph, without explicit personalization. The authors prove stability criteria and $O(\varepsilon)$ error bounds for approximately compatible systems, and report experiments on synthetic data, MNIST, and CIFAR-10. For ML researchers working on federated or decentralized training, the practical question is how realistic the required automorphism-compatibility assumptions are for real networks and heterogeneous data.
Data-Topology Compatibility Enables Personalization in Decentralized Learning
arXiv:2610.15126v1 Announce Type: cross Abstract: Decentralized learning must balance collaboration with personalization across heterogeneous agents. This paper studies when cooperation can support statistically coherent groups of personalized models. We show that data-topology compatibility enables robust orbit-wise personalization: when local objectives and communication topology have (approximately) compatible automorphisms, agents within an orbit share (approximately) equal limiting distributions, while different orbits remain distinct without explicit personalization. This creates a new intermediate regime between global consensus and independent local learning. Under exact data-topology compatibility, as a reference system, we decompose the stochastic learning dynamics into orbit-level and transverse components, derive a necessary and sufficient stability criterion for distributional synchronization, and characterize the orbit-structured stationary law. Modelling an approximate data-topology compatible system as an $\varepsilon$-perturbed reference system, we prove the introduced errors have uniform-in-time and stationary $O(\varepsilon)$ bounds in 2-Wasserstein distance. We also prove sufficient conditions for lower stationary average excess risk than both independent local training and the best single shared parameter. Experiments on synthetic data, MNIST, and CIFAR-10 are in full agreement with the theory. The code is available at https://github.com/Veronicaaa1/orbit-personalization.
Abstract:Decentralized learning must balance collaboration with personalization across heterogeneous agents. This paper studies when cooperation can support statistically coherent groups of personalized models. We show that data-topology compatibility enables robust orbit-wise personalization: when local objectives and communication topology have (approximately) compatible automorphisms, agents within an orbit share (approximately) equal limiting distributions, while different orbits remain distinct without explicit personalization. This creates a new intermediate regime between global consensus and independent local learning. Under exact data-topology compatibility, as a reference system, we decompose the stochastic learning dynamics into orbit-level and transverse components, derive a necessary and sufficient stability criterion for distributional synchronization, and characterize the orbit-structured stationary law. Modelling an approximate data-topology compatible system as an $\varepsilon$-perturbed reference system, we prove the introduced errors have uniform-in-time and stationary $O(\varepsilon)$ bounds in 2-Wasserstein distance. We also prove sufficient conditions for lower stationary average excess risk than both independent local training and the best single shared parameter. Experiments on synthetic data, MNIST, and CIFAR-10 are in full agreement with the theory. The code is available at this https URL.
Machine Learning (stat.ML); Machine Learning (cs.LG)
arXiv:2610.15126 [stat.ML]
(or arXiv:2610.15126v1 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2610.15126 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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