arXiv:2609.09676v1 Announce Type: cross Abstract: Muon replaces matrix momentum with an approximately orthogonal polar direction, but its geometry depends on the matrix representation. For convolutio...
Muon replaces matrix momentum with an approximately orthogonal polar direction, but its geometry depends on the matrix representation. For convolution, standard unfolding describes a local patch map r...
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:2512. 04632v2 Announce Type: replace Abstract: Orthogonality-based optimizers, such as Muon, have recently shown strong performance across large-scale training and community-driven efficiency challenges.
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: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...
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.
arXiv:2606. 00371v1 Announce Type: new Abstract: Muon optimizers improve neural-network training by replacing ill-conditioned momentum updates with approximately semi-orthogonal updates.
arXiv:2606. 25975v1 Announce Type: new Abstract: Common first-order optimizers, such as Adam, implicitly treat each parameter block as an unstructured vector, which disregards the multilinear weight structure present in many modern machine learning models.
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.
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.
arXiv:2605. 11396v2 Announce Type: replace Abstract: The Muon optimizer has emerged as a compelling alternative to Adam for training large language models, achieving remarkable computational savings through gradient orthogonalization.