Schatten-$\infty$ based optimizers such as Muon have shown promising empirical performance, but there remains seemingly conflicting observations regarding whether they are beneficial. We resolve this conflict by showing that the conclusion is regime dependent. Even when the objective is smooth in the Schatten-$\infty$ geometry, smaller Schatten-$p$ geometries can be optimal, specifically in the low-dimensional regime, which we show includes Chinchilla scaling. This conclusion follows from a new noise-robust acceleration result for the SODA framework for $p>2$. The same analysis explains why Muon-like methods do not require warmup, why they naturally favor large batches, and yields a batch size scaling rule for arbitrary $p$.

翻译：基于Schatten-$\infty$范数的优化器（如Muon）在实证中展现出令人鼓舞的性能，但关于其是否真正有效仍存在看似矛盾的观测结果。我们通过证明结论具有机制依赖性来解决这一矛盾。即使目标函数在Schatten-$\infty$几何下是光滑的，更小的Schatten-$p$几何仍可能达到最优，特别是在低维机制中——我们证明该机制包含Chinchilla缩放定律。该结论源自SODA框架在$p>2$条件下新的噪声鲁棒加速结果。同一分析解释了为何类Muon方法无需预热、天然偏好大批量，并推导出任意$p$下的批量大小缩放规则。