Hyperball May Not Be a Free Lunch
Not on the map. Why?
Abstract (arXiv)
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 performance. Thus, maintaining a constant angular velocity does not eliminate the learning-rate-scheduling problem; careful scheduling remains essential to realizing the potential of Hyperball-style optimizers. Our code is publicly available at https://github.com/mangocrazz/hyperball-may-not-be-a-free-lunch.
Key facts. Cited by 3 papers in the map (data through 7 Oct 2026). Builds on 22 of its 27 references that are in the map. First posted on arXiv 24 Jul 2026.
Builds on
22 of 27 references.
References · 22
- 2018Group Normalizationcited by 1,514
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- 2016Weight Normalization: A Simple Reparameterization to Accelerate Training of Deep Neural Networkscited by 541
- 2024The Road Less Scheduledcited by 158
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- 2021Label Noise SGD Provably Prefers Flat Global Minimizerscited by 121
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- 2024Understanding Optimization in Deep Learning with Central Flowscited by 57
- 2025Power Lines: Scaling Laws for Weight Decay and Batch Size in LLM Pre-trainingcited by 41
10 more
- 2024u-μP: The Unit-Scaled Maximal Update Parametrizationcited by 29
- 2026Controlled LLM Training on Spectral Spherecited by 28
- 2025Weight Decay may matter more than muP for Learning Rate Transfer in Practicecited by 20
- 2026Fantastic Pretraining Optimizers and Where to Find Them II: Hyperball Optimizationcited by 12
- 2026RMNP: Row-Momentum Normalized Preconditioning for Scalable Matrix-Based Optimizationcited by 11
- 2026ScheduleFree+: Scaling Learning-Rate-Free & Schedule-Free Learning to Large Language Modelscited by 6
- 2026Optimal Learning Rate Schedules under Functional Scaling Laws: Power Decay and Warmup-Stable-Decaycited by 4
- 2026Fast Catch-Up, Late Switching: Optimal Batch Size Scheduling via Functional Scaling Lawscited by 4
- 2026Nexus: Same Pretraining Loss, Better Downstream Generalization via Common Minimacited by 4
- 2026OptMuon: Closed-Loop Orthogonalized Momentum Methods for Stochastic Optimization with Zero-Noise Optimalitycited by 1
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Citing papers per year: 2026: 3.
3 papers.