PolarAdamW: Disentangling Spectral Control and Schur Gauge-Equivariance in Matrix Optimisation
Muon's matrix-level update couples two distinct effects: spectral control via a polar map, and equivariance under orthogonal changes of multiplicity-space basis (Schur gauge-equivariance). We separate them with PolarAdamW, a controlled hybrid that preserves Muon's polar spectral-norm control but breaks the gauge-equivariance, since AdamW's coordinatewise preconditioner is basis-dependent. Algorithmically, PolarAdamW applies Muon's Newton-Schulz polar map to AdamW's preconditioned direction rather than to raw momentum, at per-iteration wall-time comparable to Muon. We prove that Muon's polar step is Schur gauge-equivariant on multiplicity matrices while AdamW's coordinatewise step is not. ...
| Authors | Haozhou Zhang |
|---|---|
| Primary category | cs.LG |
| Published | 8 May 2026 |
| Last updated at source | 8 May 2026 |
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