Prior work on implicit bias shows that gradient descent selects the Frobenius max-margin direction on linearly separable problems,
whereas Fan et al. (2025) established spectral max-margin behavior for spectral-norm steepest descent methods, including idealized Muon with exact orthogonalization.
In practice, however, Muon approximates orthogonalization using finite Newton-Schulz iterations and a positive stabilization constant.
We prove that these implementation choices cause Muon to select the Frobenius rather than the spectral max-margin direction in linear classification with separable data.
We also establish this behavior for both exponential moving average and Nesterov-style momentum.
The key technical ingredient is a summable perturbation argument showing that the resulting parameter trajectories remain at bounded distance from those of gradient descent.
We then disentangle the roles of Newton-Schulz iteration count, the stabilization constant, and the Newton-Schulz polynomial, revealing distinct asymptotic behaviors, including directions that achieve neither Frobenius nor spectral max-margin.
Finally, we show that Muon can recover its spectral max-margin bias with Newton-Schulz iterations growing logarithmically over training.