Hyperball May Not Be a Free Lunch

作者：Y H Xiao, Jialong Sun, Gao Z, Zeming Wei, Chutian Wang, Ran Tao, Jiaye Teng, Bryan Dai · 发表于：arXiv (Cornell University) · 年份：2026 · DOI：10.48550/arxiv.2607.22444 · 研究领域：Stochastic Gradient Optimization Techniques、Machine Learning in Materials Science、Advanced Neural Network Applications

For scale-invariant deep networks, Hyperball-style optimizers have shown strong performance in large-scale training by fixing the norms of matrix-valued parameters and normalizing updates. However, the source of their advantage remains unclear. Starting from the angular displacement between consecutive parameter states, we derive an angular effective learning rate that accounts for the parameter-update angle, parameter norm, and update norm. We also show that the conventional norm-based measure is a special case under parameter-update orthogonality. We then decompose optimizer updates into radial and tangential components and analyze how radial updates affect one-step angular displacement. Under the training configurations considered, numerical results show that the radial component has only a limited direct effect on the angular effective learning rate. It therefore cannot explain why MuonH converges more slowly than MuonWD early in training but overtakes it later. To further isolate the underlying mechanism, we devise a heuristic experiment that modifies only the learning-rate schedule so that the dynamics of each optimizer reproduce those of the other. The results suggest that their main difference stems from the evolution of the effective step size rather than an intrinsically superior update direction induced by Hyperball. Our pretraining experiments further show that more aggressive learning-rate decay can accelerate MuonH early in training but may impair its later perf...