arXiv:2609.35297v3 Announce Type: replace
Abstract: Pretraining a language model takes enormous compute, and the right optimizer can save a good part of it. Muon's spectral step gives a stronger dire...
By Arman Bolatov, Artem Riabinin, Nikita Kornilov, Andrey Veprikov, Samuel Horv\'ath, Martin Tak\'a\v{c}, Aleksandr Beznosikov
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.
By Yinze Hu, Hongjun Xiang, Xingao Gong, Hongyu Yu
arXiv:2607. 29674v1 Announce Type: cross Abstract: SignMuon compresses the Muon update to one bit per parameter by taking its elementwise sign, providing the most direct way to run a matrix-aware optimizer under an extremely low communication budget.
By Maria Smirnova, Alexey Kravatskiy
arXiv:2605. 17109v3 Announce Type: replace-cross Abstract: In recent years, Muon has emerged as the dominant method for training large language models, and transformers more broadly.
By Fangzhou Wu, Rikhav Shah, Sandeep Silwal, Qiuyi Zhang
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 evaluates Muon, a data‑efficient optimizer, against AdamW on a single A100 GPU across 17.7M‑ and 33.6M‑parameter causal language models and a 2.70M‑parameter vision transformer. Using a 23‑candidate development budget and five matched seeds, Muon consistently reduces validation‑loss area under the curve by 3.25–22.44% and lowers test loss in all 20 primary pairs, while AdamW outperforms Muon on language tasks by 3.38–6.96%. Muon’s full optimizer call consumes 56.1–69.6% of training time versus 6.1–9.1% for AdamW, and it shows higher expected regret under restricted candidate pools, indicating trade‑offs between data efficiency and computational cost for small models with reference kernels.
By Mushir Akhtar, Hazem N. Nounou, Majdi Mansouri, Mohamed N. Nounou