When and Why Grouping Attention Heads Accelerates Muon Optimization

Muon orthogonalizes matrix updates, but multi-head attention naturally operates at the level of heads. This granularity mismatch raises the question of whether Muon should be applied to the full attention projection, to individual heads, or to intermediate head groups. We study this question through a one-step descent comparison between full-matrix Muon and group-wise Muon. Our analysis reveals a trade-off between the $\mathbf{\text{group-wise}}\mathbf{\text{whitening}}\mathbf{\text{gain}}$ from group-wise updates and the $\mathbf{\text{grouping-induced}}\mathbf{\text{norm}}\mathbf{\text{cost}}$, an additional update-norm cost caused by replacing full-matrix whitening with group-wise whitening. Motivated by this trade-off, we propose $\mathbf{\text{Group}}\mathbf{\text{Muon}}$, which treats head group size and grouping rule as optimizer hyperparameters. On GPT-2 Small trained on FineWeb, appropriate grouping improves validation loss over both full-QKV Muon and fully head-wise MuonSplit.