Related papers: HTMuon: Improving Muon via Heavy-Tailed Spectral C…

Orthogonalized-momentum optimizers such as Muon improve transformer training by approximately whitening/orthogonalizing matrix-valued momentum updates via a short polar-decomposition iteration. However, polar-factor approximations typically…

The Muon optimizer has received considerable attention for its strong performance in training large language models, yet the design principle behind its matrix-gradient orthogonalization remains largely elusive. In this paper, we introduce…

In recent years, Muon has emerged as the dominant method for training large language models, and transformers more broadly. The essential difference, when compared to standard gradient descent methods, is to replace the usual update matrix…

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…

In large-scale optimization, the cheapness and effectiveness of update steps are the most crucial factors for a successful optimizer. Sign-based optimizers like Lion or Signum produce cheap per-step updates, whereas Muon's spectral…

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 choice of optimizer significantly impacts the training efficiency and computational costs of large language models (LLMs). Recently, the Muon optimizer has demonstrated promising results by orthogonalizing parameter updates, improving…

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…

Physics-informed neural networks and neural operators often suffer from severe optimization difficulties caused by ill-conditioned gradients, multi-scale spectral behavior, and stiffness induced by physical constraints. Recently, the Muon…

Neural network (NN) training is inherently a large-scale matrix optimization problem, yet the matrix structure of NN parameters has long been overlooked. Recently, the optimizer Muon \citep{jordanmuon}, which explicitly exploits this…

Given two or more Deep Neural Networks (DNNs) with the same or similar architectures, and trained on the same dataset, but trained with different solvers, parameters, hyper-parameters, regularization, etc., can we predict which DNN will…

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…

Most first-order optimizers treat matrix-valued parameters as vectors, ignoring the intrinsic geometry of hidden-layer weights in neural networks. Muon addresses this mismatch by updating along the polar factor of a momentum matrix, but its…

Muon updates matrix parameters via the matrix sign of the gradient and has shown strong empirical gains, yet its dynamics and scaling behavior remain unclear in theory. We study Muon in a linear associative memory model with softmax…

Recently, a new optimization method based on the linear minimization oracle (LMO), called Muon, has been attracting increasing attention since it can train neural networks faster than existing adaptive optimization methods, such as Adam. In…

While spectral-based optimizers like Muon operate directly on the spectrum of updates, standard adaptive methods such as AdamW do not account for the spectral structure of weights and gradients, leaving them vulnerable to two empirical…

Learning rate configuration is a fundamental aspect of modern deep learning. The prevailing practice of applying a uniform learning rate across all layers overlooks the structural heterogeneity of Transformers, potentially limiting their…

Recent theoretical and empirical successes in deep learning, including the celebrated neural scaling laws, are punctuated by the observation that many objects of interest tend to exhibit some form of heavy-tailed or power law behavior. In…

Large language models (LLMs) have achieved remarkable performance on various natural language tasks. However, they are trained on static corpora and their knowledge can become outdated quickly in the fast-changing world. This motivates the…

The use of momentum in stochastic optimization algorithms has shown empirical success across a range of machine learning tasks. Recently, a new class of stochastic momentum algorithms has emerged within the Linear Minimization Oracle (LMO)…