Fixed-stabilization spectral recovery for the quintic Newton–Schulz polynomial

Determine whether spectral max-margin bias can be recovered for the quintic Newton–Schulz polynomial when the stabilization parameter is fixed and positive.

Background

The paper studies practical Muon for separable linear classification, where matrix updates are approximated by a finite number of Newton–Schulz polynomial iterations applied after stabilization. For a fixed finite iteration count and positive stabilization, the paper proves that practical Muon selects the Frobenius max-margin direction rather than the spectral max-margin direction.

The paper establishes spectral max-margin recovery for the cubic Newton–Schulz polynomial using iteration counts that grow logarithmically with training time while keeping the stabilization parameter fixed. For the quintic polynomial used in practical Muon, however, increasing the iteration count alone does not generally converge to the exact polar factor. The paper instead proves recovery under a gradient-dependent, randomized stabilization scheme, and explicitly leaves unresolved whether recovery is possible with a fixed positive stabilization parameter.