arXiv:2602. 05725v3 Announce Type: replace Abstract: Muon updates matrix parameters via the matrix sign of the gradient and has shown strong empirical gains, yet its dynamics and scaling behavior remain unclear in theory.
arXiv:2607. 13246v1 Announce Type: cross Abstract: Muon has recently emerged as a strong optimizer for large-scale deep learning, where it reshapes gradient updates through approximate orthogonalization and has been reported to outperform Adam and AdamW in large language model training.
Muon has recently emerged as a strong optimizer for large-scale deep learning, where it reshapes gradient updates through approximate orthogonalization and has been reported to outperform Adam and AdamW in large language model training. Its empirical success has motivated a growing body of theoretical work that interprets Muon as steepest descent under the spectral norm.
arXiv:2606. 08388v1 Announce Type: new Abstract: Muon replaces a matrix gradient $G=U\Sigma V^\top$ by its polar factor $UV^\top$.
AF‑Muon is an AdamW‑free extension of the Muon optimizer that retains Muon’s matrix update for hidden weights while applying a support‑aware finite‑cap linear minimization oracle to tied vocabulary tables and an RMS‑normalized update for one‑dimensional auxiliary parameters. This design eliminates second‑moment state, reducing optimizer‑state memory by about 20% compared to Hybrid Muon. Across nine tied‑token settings—including decoder‑only language models, T5‑style encoder‑decoders, and ImageGPT‑style variants—AF‑Muon consistently improves mean validation loss and perplexity over both Hybrid Muon and a SCION‑style Sign endpoint, with robust gains confirmed by long‑horizon runs and hyperparameter studies.
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:2607. 19771v1 Announce Type: cross Abstract: Muon and related matrix-sign optimizers are increasingly used to pre-train large language models, but their effect on the internal geometry of individual weight matrices is not well understood.
arXiv:2603. 00742v2 Announce Type: replace Abstract: While Adam has long been the ubiquitous default optimizer for deep neural networks, Muon has recently seen rapid adoption due to its superior training speed.
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.
arXiv:2606. 27153v1 Announce Type: cross Abstract: Matrix-orthogonalization-based optimizers, exemplified by Muon, have demonstrated strong convergence behavior across a wide range of modern deep learning workloads.
arXiv:2604. 09967v2 Announce Type: replace-cross Abstract: Muon has emerged as a promising optimizer for large-scale foundation model pre-training by exploiting the matrix structure of neural network updates through iterative orthogonalization.
arXiv:2606. 30509v1 Announce Type: new Abstract: Matrix factorization (i.
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.