MuonIO: Principled Norm-Aware Descent for Embedding Tables and Language Model Heads
The Muon optimizer derives its update rule for hidden linear layers by solving a local linearization of the loss penalized by the spectral norm, motivated by an RMS-stability argument for dense linear layers. Standard Muon implementations, however, exclude the input (embedding table) and output (language model head) layers from this principled treatment, for which they use AdamW instead. We present MuonIO, a single Muon-style update for both of these layers. For the language model head $\mathbf{L} \in \mathbb{R}^{V \times d}$, we motivate the use of the $2\to\infty$ operator norm, due to the Lipschitz continuity of the softmax output geometry, while for the embedding table $\mathbf{E} \in...
| Authors | Linkai Ma, Xinyu Luo, Mengbo Wang et al. |
|---|---|
| Primary category | cs.LG |
| Published | 2 October 2026 |
| Last updated at source | 2 October 2026 |
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