Federated Learning (FL) enables collaborative training of complex models across large-scale networks of edge devices while keeping data localized. However, data heterogeneity and high communication overhead remain fundamental bottlenecks. Recently, the spectral optimizer Muon has demonstrated remarkable advantages over Adam-like methods in centralized training. While recent efforts have attempted to transition Muon to FL, naive integrations into traditional architectures fail short. Crucially, existing approaches do not support the the communication compression and local steps --- the two foundational techniques of efficient FL. To address this limitation, we introduce a novel spectral FL method that utilizes both local updates and compression. We establish rigorous convergence guarantees under mild assumptions, notably eliminating the restrictive bounded heterogeneity condition. Empirical evaluation on standard image classification benchmarks validates the efficiency and robustness of our approach.

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