NBCLT: Parameter-Agnostic Norm-Ball Clipped Local Training

Distributed learning with multiple local steps reduces communication but aggravates drift, and its practical performance often depends on carefully tuned hyperparameters under heterogeneous data and communication constraints. At the same time, recent Muon-type optimizers suggest that constraining updates within suitable norm balls provides a simple way to encode geometric priors into optimization. In this work, we propose NBCLT (Norm-Ball Clipped Local Training), a unified local-training framework for both federated learning (FL) and decentralized learning (DL). NBCLT combines a predict-and-correct local update rule with Euclidean projections onto flexible norm balls, thereby performing norm-ball clipping under flexible update geometries while retaining a simple implementation. In FL, its problem-parameter-free regime achieves $\mathcal{O}(ε^{-2})$ communication complexity and $\mathcal{O}(ε^{-4})$ total stochastic-gradient complexity without assuming bounded heterogeneity, when the per-round local-computation budget is scaled appropriately with the horizon. In DL, NBCLT is compatible with both static and time-varying mixing and preserves these accuracy exponents under the analogous computation scaling, up to fixed network constants and an additive initialization cost.