Aligning Theory with Practice for Muon-type Optimizers: A Layer-wise Framework

Recent developments in deep learning optimization have brought about radically new algorithms based on the Linear Minimization Oracle (LMO) framework, such as 𝖬𝗎𝗈𝗇 and 𝖲𝖼𝗂𝗈𝗇. After over a decade of 𝖠𝖽𝖺𝗆's dominance, these LMO-based methods are emerging as viable replacements, offering several practical advantages such as improved memory efficiency, better hyperparameter transferability, and most importantly, superior empirical performance on large-scale tasks, including LLM training. However, a significant gap remains between their practical use and our current theoretical understanding: prior analyses (1) overlook the layer-wise LMO application of these optimizers in practice, and (2) rely on an unrealistic smoothness assumption, leading to impractically small stepsizes. To address both, we propose a new LMO-based method called 𝖦𝗅𝗎𝗈𝗇, capturing prior theoretically analyzed methods as special cases, and introduce a new refined generalized smoothness model that captures the layer-wise geometry of neural networks, matches the layer-wise practical implementation of 𝖬𝗎𝗈𝗇 and 𝖲𝖼𝗂𝗈𝗇, and leads to convergence guarantees with strong practical predictive power. Unlike prior results, our theoretical stepsizes closely match the fine-tuned values reported by Pethick et al. (2025). Our experiments with NanoGPT and CNN confirm that our assumption holds along the optimization trajectory, ultimately closing the gap between theory and practice.