Hierarchical Muon: Tiled Newton-Schulz Updates for Efficient Muon Optimization
Muon-type optimizers construct update directions for dense neural-network weights by applying a finite Newton-Schulz map to momentum-gradient matrices. For an $H \times W$ matrix, with $r=\min\{H,W\}$ and $s=\max\{H,W\}$, $K$ steps of the full-matrix Newton-Schulz update require $O(r^2 s K)$ work and couple all rows and columns through repeated Gram matrix products. We introduce Hierarchical Muon (HiMuon), a tiled Newton-Schulz scheme for Muon-type optimization. HiMuon partitions each momentum-gradient matrix into $T \times T$ tiles, applies the same finite Newton-Schulz map independently to each tile, and reassembles the results. For finite $T$ below the matrix dimensions, HiMuon defines...
| Authors | Ziyuan Tang, Tianshi Xu, Yousef Saad et al. |
|---|---|
| Primary category | math.NA |
| Published | 25 June 2026 |
| Last updated at source | 25 June 2026 |
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