Related papers: The Newton-Muon Optimizer
Large Language Models (LLMs) achieve competitive performance across diverse natural language processing (NLP) tasks, yet pretraining is computationally demanding, making optimizer efficiency an important practical consideration. Muon…
The Muon optimizer has demonstrated strong empirical performance in pre-training large language models by performing matrix-level gradient (or momentum) orthogonalization in each layer independently. In this work, we propose TEON, a…
Large models recently are widely applied in artificial intelligence, so efficient training of large models has received widespread attention. More recently, a useful Muon optimizer is specifically designed for matrix-structured parameters…
The Muon optimizer, a matrix-structured algorithm that leverages spectral orthogonalization of gradients, is a milestone in the pretraining of large language models. However, the underlying mechanisms of Muon -- particularly the role of…
Muon orthogonalizes the momentum buffer before each update, replacing its singular values with ones via Newton-Schulz iterations. This simple change lets Muon tolerate far larger learning rates and converge faster than other optimizers, but…
The Muon optimizer has demonstrated remarkable empirical success in handling matrix-structured parameters for training neural networks. However, a significant gap remains between its practical performance and theoretical understanding.…
Muon improves neural-network training by orthogonalizing matrix-valued updates, but it leaves each layer's update magnitude controlled mostly by a global learning rate. We introduce OrScale, a trust-ratio extension of Muon built on a simple…
The rapid progress of large language models (LLMs) is increasingly constrained by memory and deployment costs, motivating compression methods for practical deployment. Many state-of-the-art compression pipelines leverage the low-rank…
The Muon optimizer, based on matrix orthogonalization, has recently shown faster convergence and better computational efficiency over AdamW in LLM pre-training. However, the memory overhead of maintaining high-precision optimizer states…
Orthonormalized updates accelerate training, improve stability, and enable robust hyperparameter transfer, but existing methods like Muon rely on dense matrix operations that clash with sharded weights in large-scale LLM training, causing…
Muon has recently emerged as a strong optimizer for large language model pre-training, orthogonalizing the momentum matrix via Newton--Schulz polar iterations. A natural intuition is that polar iterations, by flattening the singular…
Optimization with matrix gradient orthogonalization has recently demonstrated impressive results in the training of deep neural networks (Jordan et al., 2024; Liu et al., 2025). In this paper, we provide a theoretical analysis of this…
Efficient stochastic optimization typically integrates an update direction that performs well in the deterministic regime with a mechanism adapting to stochastic perturbations. While Adam uses adaptive moment estimates to promote stability,…
Gradient orthogonalization is a simple strategy that shows great utility in speeding up gradient descent. The Muon optimizer (Jordan, Jin, et al., 2024) combines gradient orthogonalization with first-order momentum and achieves significant…
Orthogonalized-update optimizers such as Muon improve training of matrix-valued parameters, but existing extensions typically either rescale updates after orthogonalization or use heavier whitening-based preconditioners before it. We…
Zeroth-order (ZO) optimization provides a gradient-free alternative to first-order (FO) methods by estimating gradients via finite differences of function evaluations, and has recently emerged as a memory-efficient paradigm for fine-tuning…
We analyze Muon as originally proposed and used in practice -- using the momentum orthogonalization with a few Newton-Schulz steps. The prior theoretical results replace this key step in Muon with an exact SVD-based polar factor. We prove…
Recent advances in spectral optimization, notably Muon, have demonstrated that constraining update steps to the Stiefel manifold can significantly accelerate training and improve generalization. However, Muon implicitly assumes an isotropic…
The Muon optimizer has recently attracted attention due to its orthogonalized first-order updates, and a deeper theoretical understanding of its convergence behavior is essential for guiding practical applications; however, existing…
In this paper, we propose DeMuon, a method for decentralized matrix optimization over a given communication topology. DeMuon incorporates matrix orthogonalization via Newton-Schulz iterations-a technique inherited from its centralized…