DynMuon: A Dynamic Spectral Shaping View of Muon

Muon is an increasingly widely used optimizer that replaces a gradient $G=USV^\top$ with its polar factor $UV^\top$, thereby flattening the singular spectrum. However, full flattening discards singular-value information that may matter for adaptation. We introduce Muon$^p$, a Muon-style optimizer that instead uses fractional spectral-power updates $US^pV^\top$ for rational $p\in(0,1)$, interpolating between Muon and gradient descent. To make it practical, we prove that fractional spectral powers cannot be computed by any fixed univariate polynomial iteration, and furthermore derive low-degree odd bivariate recurrences that approximate $US^pV^\top$ using only matrix multiplications, preserving Muon's matrix-multiplication-only structure and compute complexity. We show that Muon$^p$ maximizes the linear improvement in loss under the Schatten $q$-norm for $q=1+\frac{1}{p}$. Empirically, Muon$^p$ is especially effective for finetuning: on billion-scale models, Muon$^p$ improves validation perplexity and downstream task performance. We further analyze when Muon$^p$ is less suitable, through the lens of spectral geometry. Our results reveal important insights on when preserving the singular spectrum can bring significant gains, and introduce a principled way to achieve them.

Muon has recently demonstrated strong empirical performance in large language model training, but the theoretical role of momentum in Muon remains unclear. Existing analyses of Muon either remove momentum to study spectral updates in isolation, or retain momentum without explaining why it improves empirical performance. Our work bridges this gap by showing momentum in Muon acts as a spectral filter. Under a structured signal-plus-perturbation gradient model, we prove that momentum suppresses perturbations while preserving the dominant signal, thereby enlarging the spectral gap between them. This enlarged gap stabilizes the singular subspaces of the matrix passed to Muon's orthogonalization step, making the resulting update more reliable. We further show that applying momentum before orthogonalization achieves provably stronger alignment with the signal component of the gradient than either reversing this order or simply removing momentum. Experiments across diverse tasks, including LLM pretraining, support our theoretical analysis. More broadly, our theory offers a starting point for understanding the benefits of momentum in other matrix-based optimizers.

Muon has emerged as a highly effective optimizer for large language model training, often achieving superior convergence and performance compared with the widely adopted Adam and AdamW optimizers. Nevertheless, Muon is prone to training instability due to its spectral flattening, manifested by loss spikes and unbounded growth of model weights. Existing approaches primarily rely on weight or attention-logit clipping, which require architecture-specific modifications and do not directly address instability across all model components. We propose MomentUm SpEctral Clipping (Musec), which replaces Muon's spectral flattening with spectral clipping: rather than setting all singular values of the momentum matrix to approximately one, Musec clips singular values that exceed a threshold while preserving the underlying spectral structure of the momentum. Our strategy provides an optimizer-level, architecture-agnostic mechanism for stabilizing Muon training. Theoretically, we establish convergence guarantees for Musec in nonconvex nonsmooth stochastic optimization. Practically, we develop Soft Musec, an efficient implementation that uses a smooth spectral saturation function approximated by coupled Newton-Schulz iterations. Empirically, Soft Musec consistently improves training stability over existing Muon variants across a wide range of learning rates and model sizes, remaining stable in settings where existing Muon variants diverge while matching their performance under well-tuned configurations.