Muon Sublates the Edge of Stability in LLM Pretraining

The success of large language models (LLMs) has been accompanied by continued growth in model size and pretraining costs. Muon offers high accuracy and training efficiency in LLM pretraining. Recent work introduces row-wise normalization into Muon to balance update magnitudes and improve pretraining performance. However, row-wise normalization alone cannot accommodate different imbalance patterns in update matrices. In this paper, we propose an improved Muon optimizer, called \underline{m}atrix-\underline{eq}uilibrating Muon~(MeqMuon), for LLM pretraining. MeqMuon balances both row and column magnitudes through normalization that can be automatically tailored to different imbalance patterns without manual intervention. Moreover, MeqMuon eliminates the need to store AdamW's second-moment estimates, reducing optimizer-state memory usage. Empirical results demonstrate that MeqMuon achieves better convergence performance than AdamW, Muon, and other baselines in LLM pretraining.

Muon has recently emerged as a promising alternative to AdamW for language model pretraining by orthogonalizing momentum matrices using Newton-Schulz iterations. Although Muon mitigates gradient anisotropy, it does not explicitly account for the curvature geometry of the loss landscape and may therefore remain sensitive to curvature anisotropy. We bridge this gap by proposing MALT (Muon Augmented by Lightweight Two-sided Preconditioning), which uses lightweight diagonal preconditioners to reduce the sensitivity of Muon to curvature anisotropy. Specifically, MALT uses two-sided diagonal preconditioners with low memory and computational overhead to approximately capture the curvature geometry of the loss landscape. It orthogonalizes the preconditioned momentum using Newton-Schulz iterations and maps the result back to define the update direction, while norm grafting controls the update magnitude. To improve the robustness of MALT to stochastic gradient noise, we further propose MALTER (MALT with Adaptive stEpsize Rescaling). Convergence guarantees are provided for MALT in the stochastic non-convex setting. Experiments on GPT-2 Small, Medium, and Large pretraining show that the proposed methods outperform Muon while maintaining nearly the same memory footprint and wall-clock time.

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. Our code is available at https://github.com/fzwark/DynMuon.