Orth-Dion: Eliminating Geometric Mismatch in Distributed Low-Rank Spectral Optimization
Low-rank gradient compression reduces communication in distributed training by representing updates with rank-$r$ factors. Dion is a recent method that approximates Muon, a spectral optimizer that orthogonalizes momentum, using one step of power iteration followed by column normalization (rescaling each column of the right factor to unit length). This makes it compatible with fully sharded data parallel training, but it converges more slowly than full-rank spectral methods. We show that this gap is geometric: column normalization does not yield the rank-$r$ polar factor that Muon implicitly targets, so the resulting direction violates the dual-norm constraint of the low-rank spectral geom...
| Authors | Tatsuhiro Nakamori, Laura Gomezjurado Gonzalez, Ganesh Talluri et al. |
|---|---|
| Primary category | cs.LG |
| Published | 7 May 2026 |
| Last updated at source | 7 May 2026 |
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