Cost-free Spectral Estimation for Adaptive Newton--Schulz in Matrix Optimizers
Matrix optimizers such as Muon transform each momentum matrix through an approximate orthogonalization, typically implemented by a small number of Newton-Schulz matrix multiplications. The quality and cost of this approximation depend strongly on the singular-value spectrum of its input, yet existing implementations use the same fixed polynomial routine for every layer and throughout training. We show that this uniform treatment is unnecessary: the computations in the Newton-Schulz method already reveal enough information to make the method adaptive. The Gram matrices formed inside Newton-Schulz iterations yield spectral moments through inexpensive scalar reductions, requiring no addition...
| Authors | Kristi Topollai, Anna Choromanska |
|---|---|
| Primary category | cs.LG |
| Published | 27 September 2026 |
| Last updated at source | 27 September 2026 |
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