The Implicit Bias of Adam and Muon on Smooth Homogeneous Neural Networks

We study the implicit bias of momentum-based optimizers on smooth homogeneous models. We show that $\mathit{\text{momentum}}\mathit{\text{steepest}}\mathit{\text{descent}}$ algorithms like Muon (spectral norm), MomentumGD (${\mathrm{\ell }}_2$ norm), and Signum (${\mathrm{\ell }}_{\mathrm{\infty }}$ norm) are $\mathit{\text{approximate}}$ steepest descent trajectories under a decaying learning rate schedule, proving that these algorithms have a bias towards KKT points of the corresponding margin maximization problem. We extend the analysis to Adam (without the stability constant), which maximizes the ${\mathrm{\ell }}_{\mathrm{\infty }}$ margin, and to Muon-Signum and Muon-Adam, which maximize a hybrid norm. Our experiments corroborate the theory and show that the identity of the margin maximized depends on the choice of optimizer. Overall, our results extend earlier lines of work on steepest descent in homogeneous models and momentum-based optimizers in linear models.