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.

翻译：暂无翻译