Muon Operates Beyond the Edge of Stability

Niko Covic, Baraq Lipshitz, Alexander Padula, Stefan Visnjic

We study the Muon optimizer using sharpness and provide the first empirical evidence that the Flat Minima Hypothesis also applies to Muon. Further, we derive Muon's implicit preconditioner to analyze its potential Edge of Stability behavior. Experiments on CIFAR-10 show sharpness strongly correlates with generalization gap within each optimizer (Pearson $r = 0.45$-$0.69$, $p \leq 10^{-4}$), confirming the hypothesis holds for Muon similarly to Adam and SGD. However, Muon operates above the classical sharpness-based Adaptive Edge of Stability threshold that bounds diagonally-preconditioned optimizers like Adam and RMSprop, yet remains stable. Our findings suggest Muon operates in a fundamentally different regime than diagonally-adaptive methods. Defining the exact stability boundary for Muon remains an open question. All code is available at https://github.com/NikoCovic/DL-project.