Free Heavy-Tailed Lunch for Muon: A Theoretical Justification of Empirical Success
Non-Euclidean optimisation methods with matrix-valued updates, such as Muon and Scion, have recently shown strong empirical performance for training Transformer models, yet their theoretical advantages over Euclidean methods remain poorly understood. We address this gap in the heavy-tailed non-convex regime, where stochastic gradients have bounded $p$-th central moments, $p \in (1,2]$. We show that certain non-Euclidean methods achieve optimal sample complexity under stronger stationarity measures, while Euclidean methods incur additional dimension-dependent costs. As a consequence, for $m \times n$ matrices, Muon finds an $\varepsilon$-stationary point in nuclear norm within $\mathcal{O}...
| Authors | Florian Hübler, Thomas Pethick, Suvrit Sra |
|---|---|
| Primary category | math.OC |
| Published | 12 June 2026 |
| Last updated at source | 12 June 2026 |
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