The paper proposes a new update geometry for the language‑model head by treating the head and softmax as a single module and using Hilbert’s projective distance to measure functional change. It replaces the spectral norm with the Euclidean row diameter, derives a tractable RowNorm update rule, and demonstrates that RowNorm substantially reduces step diameters and Hilbert perturbations while only slightly increasing validation loss.
By Aditya Somasundaram, Charles Guille-Escuret, Alexander Moreno, Zhengzhong Liu, Eric Xing
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.
By Arash Lagzian, Paniz Halvachi, Junming Zhang, Zhouhan Lin, Dianbo Liu
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:2610.07103v1 Announce Type: cross
Abstract: Muon, an optimizer known for its efficiency, has a clear interpretation for matrix-valued updates, but convolutional kernels are stored as four-dimen...
By Thibaut Boissin (IRIT), Thomas Massena (IRIT, DTIPG - SNCF, UT3), Mathieu Serrurier (IRIT), Franck Mamalet
arXiv:2606. 30509v1 Announce Type: new Abstract: Matrix factorization (i.
By Mark Rhee, Jamie Simon, Dhruva Karkada
arXiv:2606. 25971v1 Announce Type: new Abstract: Modern neural network training relies on optimizers such as Adam and Muon which act on each weight matrix as a single object.
By Alexander H\"agele, Alejandro Hern\'andez-Cano, Atli Kosson, Martin Jaggi
Low-rank adaptation (LoRA) is the standard way to fine-tune large models, yet when its two factors are trained independently, the update ignores the geometry of the low-rank weight change it induces. We introduce LoRA-TSD, an optimizer that treats every LoRA step as a tangent vector of the fixed-rank matrix manifold and takes the spectral-norm steepest-descent step of Muon inside that tangent space, mapping the result back to the factors through a retraction native to the LoRA parametrization.
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
LoRA-TSD introduces a new optimizer for low‑rank adaptation (LoRA) that treats each update as a tangent vector on the fixed‑rank matrix manifold and applies a Muon‑style spectral‑norm steepest‑descent step within that tangent space. The method avoids costly full‑matrix operations and offers a retraction that is up to 2.8× cheaper than previous manifold approaches. The authors prove that their surrogate recovers LoRA‑Pro, identify the Riemannian gradient as the natural stationarity measure, and provide the first global convergence guarantees for both LoRA‑Pro and LoRA‑TSD, achieving superior performance across multiple benchmarks with Llama and Qwen models.
By Dmitrii Andriianov, Andrey Veprikov, Aleksandr Beznosikov
The paper introduces a data‑driven criterion that dynamically selects optimal Linear Minimization Oracle (LMO) geometries for each layer of a deep neural network, bridging the gap between SGD and Muon updates. By deriving the criterion from gradient and activation statistics via a single‑step random feature regression surrogate, the authors enable a design space that interpolates between existing optimizers, including SGD, Muon, Adam, and MuAdam. The adaptive approach incurs only about a 3% runtime overhead and demonstrates competitive or superior performance compared to the best static optimizer across three training scenarios.
By Thomas Massena (DTIPG - SNCF, IRIT), Corentin Friedrich (IRIT), Mathieu Serrurier (IRIT)
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. 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.
By Ali Parviz, Gal Mishne, Alex Cloninger