DynMuon: A Dynamic Spectral Shaping View of Muon

2026-05-16

AI Summary

This work addresses the limitation of conventional Muon optimizers, which employ a fixed spectral shaping strategy and thus struggle to accommodate the varying curvature requirements across different training phases. The authors propose DynMuon, a dynamic spectral shaping method that, for the first time, elucidates the distinct roles of positive and negative power exponents \( p \): emphasizing high-curvature directions early in training to accelerate convergence, and prioritizing low-curvature directions later to capture residual signals. Building on this insight, DynMuon introduces an adaptive scheduling strategy that dynamically adjusts the exponent applied to the singular values of the update matrix. Integrating singular value decomposition, curvature-aware analysis, and dynamic parameter scheduling, DynMuon consistently outperforms standard Muon across diverse model scales and architectures, reducing the number of training steps required to reach the same validation loss by 10.6%–26.5%.

Abstract

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 $M=UΣV^\top$ with its polar factor $UV^\top$. In this work, we consider a class of Muon-like updates, where we replace the update $M$ with $UΣ^p V^\top$ for some parameter $p$. We call this a "spectral-shaping" operation, and develop a theory of how to pick $p$ which depends on (a) local curvature of the loss function, (b) noise stemming from stochastic gradients and label noise, and (c) training stage. Our theory and experimentation reveal a previously overlooked behavior: positive $p$ helps early by emphasizing high-curvature directions and accelerating signal contraction, while mildly negative $p$ helps later by reallocating update strength toward low-curvature directions that still contain useful training signals. Building on the insight, we propose DynMuon, an efficient dynamic spectral shaping method that schedules $p$ from positive to mildly negative over training. Extensive experiments across model sizes, architectures, and training settings show that DynMuon consistently achieves lower validation loss than Muon, while requiring 10.6-26.5% fewer steps to reach the same target loss.

Problem

• spectral shaping
• Muon
• dynamic optimization
• large language models
• training efficiency

Innovation

• spectral shaping
• dynamic optimization
• Muon
• curvature-aware learning
• adaptive gradient methods