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Last-Iterate Stationarity of Muon with Finitely Many Newton-Schulz Iterations
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arXiv Machine Learning
3d ago
Last-Iterate Stationarity of Muon with Finitely Many Newton-Schulz Iterations
The paper investigates the last-iterate convergence properties of the Muon optimization algorithm, which employs a fixed number of Newton–Schulz iterations for large-scale language‑model training. It shows that without input stabilization Muon can cycle even on smooth convex quadratics, while with positive stabilization and a momentum coefficient below one, last‑iterate gradient convergence is achieved under a small constant learning rate. For stochastic gradients, the authors demonstrate that fixed learning rates can fail, but decreasing learning rates combined with increasing momentum yield almost‑sure and mean‑square convergence, and they provide counterexamples and density‑one convergence rates.
arXiv Machine Learning
Oct 2
Convergence of Practical Muon with Finite Newton-Schulz Iterations and Nesterov Momentum
arXiv:2609.39595v1 Announce Type: new Abstract: Practical Muon maintains momentum and performs a small, fixed number of Newton--Schulz iterations separately for each parameter matrix, often with a Ne...
arXiv Machine Learning
Aug 28
Muon with Finite Newton-Schulz: The Smoothing Benefit in Nonsmooth Nonconvex Optimization
The paper introduces Muon, an optimizer that uses a finite number of Newton‑Schulz iterations to approximate the polar factor for matrix‑valued parameters in large language model pretraining. It demonstrates that this finite iteration smooths the discontinuous polar map into a Lipschitz function of singular values, enabling a conversion from online learning regret to a stationarity guarantee in nonsmooth nonconvex optimization. The authors prove that a logarithmic depth in Newton‑Schulz suffices for convergence to stationary points, matching best‑known sample complexity bounds and extending the result to other spectral maps with similar smoothing properties.
arXiv Machine Learning
Aug 6
On MUON optimization: From non-convergence to an error analysis with Polar Express and the Newton-Schulz polynomial from implementations
arXiv:2608. 04607v1 Announce Type: cross Abstract: Stochastic gradient descent (SGD) optimization methods are the standard instruments for the training of deep neural networks (DNNs).
arXiv Machine Learning
Oct 15
From Spectral to Frobenius: When Muon Loses Its Spectral Implicit Bias
arXiv:2610.20112v1 Announce Type: new Abstract: Prior work on implicit bias shows that gradient descent selects the Frobenius max-margin direction on linearly separable problems, whereas Fan et al. (...
arXiv Machine Learning
Sep 11
Musec: MomentUm SpEctral Clipping for Stable Muon-type Training
Musec introduces MomentUm SpEctral Clipping, an optimizer-level, architecture‑agnostic technique that replaces Muon’s spectral flattening with selective spectral clipping to stabilize training. By clipping singular values above a threshold while preserving the momentum’s spectral structure, Musec addresses loss spikes and unbounded weight growth without requiring architecture‑specific changes. Soft Musec, an efficient implementation using smooth spectral saturation via coupled Newton‑Schulz iterations, offers convergence guarantees in nonconvex nonsmooth stochastic optimization and empirically improves stability across diverse learning rates and model sizes.