arXiv:2509. 14562v4 Announce Type: replace Abstract: Large models recently are widely applied in machine learning, so efficient training of large models has received widespread attention.
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).
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.
arXiv:2608. 12710v1 Announce Type: new Abstract: Muon, a more recently developed optimizer, is useful for matrix-wise models in AI areas.
arXiv:2507. 01598v5 Announce Type: replace Abstract: Muon, a recently proposed optimizer that leverages the inherent matrix structure of neural network parameters, has demonstrated strong empirical performance, indicating its potential as a successor to standard optimizers such as AdamW.
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:2609.07597v1 Announce Type: cross Abstract: Muon can be interpreted as optimizing a linear local objective over a spectral-norm ball. This gives a matrix-sign update that preserves the singular...
The paper introduces a physical response-and-memory model for the Muon optimizer, explaining its semi‑orthogonalized momentum update as the maximally dissipative direction under an output‑side safety budget. It treats the weight matrix as a responsive medium with internal stress, showing that momentum corresponds to accumulated stress whose relaxation occurs over multiple timescales—fast and slow. Based on this, the authors propose the Bi‑Maxwell optimizer, which uses a two‑timescale memory kernel and achieves target loss in fewer steps on a public large‑language‑model benchmark.
The paper introduces low‑rank orthogonalization, a technique that exploits the low‑rank nature of gradients in neural network training to perform matrix orthogonalization more efficiently. Building on this, the authors present low‑rank matrix‑signed gradient descent (MSGD) and a low‑rank variant of the Muon optimizer, showing through experiments that low‑rank Muon matches or surpasses vanilla Muon on GPT‑2 and LLaMA pretraining, especially for larger models. Theoretical analysis provides iteration‑complexity bounds for both low‑rank MSGD and low‑rank Muon under heavy‑tailed noise.
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:2606. 13867v1 Announce Type: new Abstract: Muon is an increasingly widely used optimizer that replaces a gradient $G=USV^\top$ with its polar factor $UV^\top$, thereby flattening the singular spectrum.
arXiv:2606. 27298v1 Announce Type: cross Abstract: We study the fundamental problem of learning a high-dimensional Gaussian truncated to an unknown halfspace.
The paper presents the first asymptotic convergence guarantees for the Muon algorithm, showing that with suitable hyperparameters the iterates satisfy ≠≠ ∥∇f(x_k)∥ → 0 and, under a global Polyak-ℒojasiewicz condition, the function values converge linearly. It reveals that Muon’s implicit regularization acts as a bounded preconditioner, framing Muon as a preconditioned Polyak heavy‑ball method and enabling a Lyapunov analysis. Building on this insight, the authors introduce Muesterov, a Nesterov‑based variant, and prove it shares the same convergence guarantees, extending the theory beyond the heavy‑ball setting; numerical experiments on a scalar cross‑entropy problem and preliminary nanoGPT simulations support the theoretical findings.