LionMuon is a new optimizer that alternates between Lion’s sign-based updates and Muon’s spectral matrix-sign updates on a fixed period P, sharing a single dual-EMA momentum buffer. This design keeps the memory footprint the same as Lion and half that of AdamW while reducing the average iteration cost compared to Muon. Experiments on 124M, 355M, and 720M models show LionMuon Pareto-dominates Muon, Lion, Signum, and AdamW across datasets and architectures, achieving lower validation loss with less compute.
By Arman Bolatov, Artem Riabinin, Nikita Kornilov, Andrey Veprikov, Samuel Horv\'ath, Martin Tak\'a\v{c}, Aleksandr Beznosikov
arXiv:2310. 15976v4 Announce Type: replace Abstract: signSGD is attractive in nonconvex optimization because it communicates sign-valued rather than full-precision gradients.
By Zhen Qin, Zhishuai Liu, Pan Xu
StoSignSGD is a new sign‑based optimization algorithm that injects structural stochasticity into the sign operator, ensuring unbiased updates. It resolves the divergence issues of traditional SignSGD on non‑smooth objectives, achieving optimal convergence rates in convex settings and improved complexity bounds in non‑convex, non‑smooth problems. Empirical results show that StoSignSGD is stable and efficient across large language model training, outperforming AdamW and SignSGD in low‑precision regimes (FP8 and FP4) and delivering speedups and accuracy gains on models ranging from OLMo2‑370M to 7B LLMs.
By Dingzhi Yu, Rui Pan, Yuxing Liu, Difan Zou, Tong Zhang
arXiv:2607. 20512v1 Announce Type: cross Abstract: The Muon optimizer reaches the grokking threshold on modular arithmetic faster than AdamW.
By Yufeng Wang
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.
By Jiachun Li
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
arXiv:2507. 12091v2 Announce Type: replace-cross Abstract: This paper presents an improved analysis for sign-based methods with momentum updates.
By Wei Jiang, Dingzhi Yu, Sifan Yang, Wenhao Yang, Zechao Li, Lijun Zhang
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
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.
By Arthur C. B. de Oliveira, Dhruv D. Jatkar, Guilherme S. Vicinansa, Eduardo D. Sontag
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...
By Hanyng Peng, Hui Wang, Yue Yu
arXiv:2609.36692v1 Announce Type: cross
Abstract: Matrix optimizers have emerged as a promising direction, with Muon standing out as a prominent design. Revisiting Muon through its full-Gram represen...
By Zixuan Gong, Zeyu Gan, Jiaye Teng, Yong Liu
arXiv:2608. 07436v1 Announce Type: new Abstract: Under the standard split, Muon gets hidden matrices and AdamW embeddings/output head.
By Ali Janati, Kaoutar El Maghraoui, Andrei Kanavalau, Anass Belfatmi