We study the implicit bias of momentum-based optimizers on homogeneous models. We first extend existing results on the implicit bias of steepest descent in homogeneous models to normalized steepest descent with an optional learning rate schedule. We then show that for smooth homogeneous models, momentum steepest descent algorithms like Muon (spectral norm), MomentumGD ($\ell_2$ norm), and Signum ($\ell_\infty$ norm) are approximate steepest descent trajectories under a decaying learning rate schedule, proving that these algorithms too 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 $\ell_\infty$ 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.

Much of the recent progress has been achieved by taking advantage of continuous relaxations of the system, which are not always available or even possible.

When n is independent ofδ our approach yields an algorithm whose sample complexityconvergesto 2n2 log 1δ asngrows.

This paper studies the problem of entropy identity testing: given sample access to a distribution p and a fully described distribution q (both discrete distributions over a domain of size k), and the promise that either p = q or |H (p) H (q)| ε, where H () denotes the Shannon entropy, a tester needs to distinguish between the two cases with high probability.

We study the problem of estimating the means of well-separated mixtures when an adversary may add arbitrary outliers.