Theoretical Analysis of Muon on Transformers for Next-Token Prediction

Muon has recently emerged as a strong optimizer for large language model pretraining by exploiting matrix structure through polar orthogonalization. However, it processes each parameter matrix independently, whereas recent tensor-aware extensions of Muon jointly update groups of eligible matrix parameters through matrix unfoldings. In contrast, we propose T-MUON, a t-SVD-based generalization that groups structurally related Transformer gradients into third-order tensors and defines the corresponding Muon-style update directly through the $M$-transform. We show that the exact T-MUON update is a steepest-descent direction under the $\star_M$-spectral norm and establish a convergence guarantee. Across GPT-2-style and LLaMA-style models from 51M to 1.3B parameters trained on FineWeb and FineWeb-Edu, head-wise T-MUON with the identity transform consistently achieves lower validation perplexity than AdamW, Muon, and variants using DFT, DCT, or DST transforms. These gains persist in additional experiments with longer context lengths and larger token budgets, although the preferred tensorization mode varies across settings.

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.

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 spectrum to all ones, should also eliminate column- and row-wise norm imbalance in the update. We show that this is not true in practice: practical polar steps can substantially amplify the imbalance. We term this the post-polar imbalanced update problem, and prove that such imbalance tightens the second-order term in a blockwise descent analysis, weakening Muon's per-step descent guarantee. Motivated by this analysis, we propose Muon+, a one-line fix that inserts a single normalization step after polar orthogonalization. Muon+ adds no optimizer state. Across pre-training experiments on GPT and LLaMA models from 60M to 7B parameters, spanning both compute-optimal budgets and extended token-to-parameter ratios up to approximately 200, Muon+ consistently outperforms Muon in terms of training and validation perplexity, leading to significant overall pre-training speedup.