A question about Adam and the norm-based perspective

[Sun2018421]

Hi authors,

I recently read your paper on (arXiv:2502.07529v2), and it's an awesome piece of work! The idea of unifying optimizers like Muon and SignSGD under a single lmo framework is super cool.
In the paper, you contrast your a priori approach of setting a norm with on-the-fly methods like Adam. This got me thinking, and I'd love to hear your thoughts on this: Is it possible to also understand the Adam optimizer from a norm-based perspective?

I understand that Adam's update step size is variable, which is different from the fixed-norm output of an lmo. However, the operation in Adam that divides by the square root of squared gradients (\sqrt(v_t)) feels like it's performing some kind of geometric normalization, which might have an intrinsic connection to norms.

More specifically, I was wondering:

1. Could Adam be viewed as using a "dynamic norm-ball," where the norm-ball itself changes at each step based on the history of gradients?
2. I found the connection you drew between weight decay and norm constraints very insightful. It immediately made me think of AdamW. You also cited the paper linking AdamW to the $l_\infty$ norm, which seems to suggest a real connection here. Do you think this could be a potential path to fit Adam into your framework?
3. Alternatively, is there a fundamental reason why this idea wouldn't work? For instance, is the key difference that prevents unification the fact that the lmo is scale-invariant, while Adam's update is sensitive to the gradient's scale?

Sorry for the long question; I'm just a student who is very interested in optimizers, and your paper has been very inspiring.

Thanks for any thoughts you might have and for the great work!

Best regards.