[Submitted on 6 Jul 2026]
Title:The Active Ingredient in Muon's Grokking
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Abstract:The Muon optimizer reaches the grokking threshold on modular arithmetic faster than AdamW. Prior work attributes this to "spectral-norm constraints plus orthogonalized momentum" but does not isolate which mechanism matters. To better understand Moun's behavior, we run multi-seed and multi-learning-rate sweeps to decompose and stress-test the effect. First, an ablation shows the speedup comes from orthogonalization (the Newton-Schulz iteration): orthogonalize-only matches full Muon, whereas spectral-only is no faster than AdamW and is unreliable, and this verdict holds across learning rates. Second, a mechanistic analysis finds that orthogonalizing optimizers reach generalization at roughly 3x lower spectral norm and, controlling for how much the embedding actually moves, settle into a lower-norm solution rather than simply perturbing the embedding less. Third, reducing the Newton-Schulz iteration count from five to one accelerates reaching the threshold but makes the grokked solution fragile, prone to transient collapse, with fragility that grows with learning rate; a single iteration is fast and stable only at small learning rate, while the canonical five iterations are the learning-rate-robust choice. We also show spectral scaling can be dropped at no measured cost. A methodological thread runs throughout: under a stability-aware metric, "faster" claims about grokking optimizers can invert, so we report both first-crossing and sustained-grok times. To support reproducibility, we release our full training and analysis code at this https URL
Subjects:
Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
Cite as: arXiv:2607.20512 [cs.LG]
(or arXiv:2607.20512v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2607.20512
arXiv-issued DOI via DataCite (pending registration)
Submission history
From: Yufeng Wang [view email] [v1] Mon, 6 Jul 2026 22:35:57 UTC (34 KB)
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