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.
By Naoki Sato, Hiroki Naganuma, Hideaki Iiduka
The paper introduces Spectrally Targeted Muon, an optimizer that orthogonalizes only singular values of update matrices above or below a threshold τ, thereby interpolating between normalized SGD and the full Muon optimizer. It uses Newton‑Schulz iteration on a shifted Gram matrix to avoid SVD, and evaluates the method on CIFAR‑10 and NanoGPT, tracking effective rank and a new alignment metric. The study finds that orthogonalizing small singular values is crucial, that shrinking large singular values keeps the parameter spectrum flat, and that AdamW differs mainly in the speed of structure formation compared to Muon.
By Vishrut Goyal, Rohan Ramkumar
The paper investigates why the orthogonal optimiser Muon outperforms Adam in large language model pretraining by analysing the spectral properties of Transformer loss landscapes. It finds that Muon’s momentum buffers exhibit an anisotropic spectral profile with a volatile head and a tolerant bulk, enabling larger effective step sizes. Building on this insight, the authors propose Spectral‑Aware Muon (SAMuon) and a lightweight variant, which adjust the bulk scaling while keeping the head unchanged, achieving 13–24 % fewer training tokens than Muon without extra FLOPs.
By Xiaodong Wu, Wenyi Yu, Chao Zhang, Philip Woodland
The paper introduces Activation-Keyed Momentum (AK‑Momentum), a momentum update that uses the input activation of a linear layer as a key to apply a delta‑rule update, allowing each direction to decay at a rate proportional to its frequency of appearance. AK‑Momentum is proven to be a valid momentum, incorporates input‑side curvature correction without matrix inversion, and clears stale directions faster than traditional exponential moving average (EMA) under both fixed and drifting optima. It can replace the momentum buffer of any optimizer, scales with width under μP, adds only 22–25% extra compute, and demonstrates significant step‑count reductions in FineWeb‑Edu pretraining and other benchmarks.
whyItMatters":"AK‑Momentum offers a principled, efficient way to adapt momentum decay to anisotropic training dynamics, improving convergence speed and stability across a range of models and datasets."
By Euijin Hong, Guannan Qu
arXiv:2606. 04058v1 Announce Type: cross Abstract: Orthonormalized update rules have rapidly become a leading choice of optimizer for training large language models, with recent open-source state-of-the-art models adopting Muon.
By Gagik Magakyan, Pablo Parrilo, Asuman Ozdaglar
The paper introduces two variance‑adaptive variants of the Muon optimizer—Muon‑NSR and Muon‑VS—for language model pretraining. Both methods incorporate gradient‑variance information into Muon’s orthogonalization process without adding extra hyperparameters, preserving its spectral normalization structure. Experiments on Llama‑style and GPT‑2 models ranging from 125 M to 1.2 B parameters show that these variants outperform well‑tuned Muon baselines and achieve up to a 1.33× step‑to‑target speedup on Llama‑1.2B.
By Jingru Li, Yibo Fan, Huan Li