AI 中文总结
研究μ子优化器在模运算中比AdamW更快达学习理解阈值的原因，通过多实验分解压力测试，发现加速源于正交化，减少迭代次数有不同影响，还表明可去掉谱缩放，且在稳定性度量下相关“更快”说法可能反转，发布代码支持重现。
AI 中文摘要
μ子优化器在模运算上比AdamW更快达到学习理解阈值。先前工作认为这归功于“谱范数约束加正交动量”，但未明确关键机制。为更好理解μ子行为，我们进行多种子和多学习率扫描来分解和压力测试效果。消融实验表明加速源于正交化；机理分析发现正交优化器在更低谱范数下达到泛化；减少牛顿 - 舒尔茨迭代次数虽能加速但使解脆弱。还表明可无代价去掉谱缩放。在稳定性感知度量下，关于学习理解优化器“更快”的说法可能反转，所以报告首次穿越和持续学习理解时间，并发布代码支持重现性。
英文摘要
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 https://github.com/louiswang524/muon-grokking-frontier
