Convergence of Practical Muon with Finite Newton-Schulz Iterations and Nesterov Momentum
Practical Muon maintains momentum and performs a small, fixed number of Newton--Schulz iterations separately for each parameter matrix, often with a Nesterov correction. We analyze these layer-wise finite-step updates jointly on a coupled nonconvex objective, rather than replacing them by exact polar factors or one global orthogonalization. Under gradient-dependent $(\mathcal L_0,\mathcal L_1,q)$-smoothness and conditionally unbiased stochastic gradients with bounded layer-wise variance, we establish an $\mathcal O(T^{-1/4})$ bound on the expected average Frobenius gradient norm. The analysis retains the Nesterov recursion and requires neither bounded stochastic gradients, symmetric noise...
| Authors | Hanyng Peng, Hui Wang, Yue Yu |
|---|---|
| Primary category | cs.LG |
| Published | 30 September 2026 |
| Last updated at source | 30 September 2026 |
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