Equivariance Breaks the Learning Rate
Equivariant networks are commonly trained with Adam, yet recent work reports that matrix structured optimizers such as Muon can perform better, with the reasons for these gains only partly understood. We identify one source of this difference inside equivariant layers. An equivariant layer learns one channel mixing matrix $W_l$ per degree $l$, which we call an irrep block, and shares it across the $2l+1$ components, giving the expanded map $W_l \otimes I_{2l+1}$. This sharing sums gradient contributions across components and can produce different update scales under SGD. Adam's entrywise normalization reduces sensitivity to gradient scale, but neither optimizer directly controls the effec...
| Authors | Andrei Manolache, Mathias Niepert |
|---|---|
| Primary category | cs.LG |
| Published | 8 September 2026 |
| Last updated at source | 30 September 2026 |
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