# Does Muon Need Fine-Grained Spectral Shaping?

arXiv: https://arxiv.org/abs/2610.07497 · HTML: https://mlsanity.com/paper/2610.07497/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 43 references that are in the map.
- First posted on arXiv 5 Oct 2026.

## Builds on (most cited first, 21 of 21)

- [The Pile: An 800GB Dataset of Diverse Text for Language Modeling](https://mlsanity.com/paper/2101.00027/) (2020)
- [Decoupled Weight Decay Regularization](https://mlsanity.com/paper/1711.05101/) (2017)
- [Muon is Scalable for LLM Training](https://mlsanity.com/paper/2502.16982/) (2025)
- [SmolLM2: When Smol Goes Big -- Data-Centric Training of a Small Language Model](https://mlsanity.com/paper/2502.02737/) (2025)
- [An Empirical Model of Large-Batch Training](https://mlsanity.com/paper/1812.06162/) (2018)
- [Old Optimizer, New Norm: An Anthology](https://mlsanity.com/paper/2409.20325/) (2024)
- [SOAP: Improving and Stabilizing Shampoo using Adam](https://mlsanity.com/paper/2409.11321/) (2024)
- [Dion: Distributed Orthonormalized Updates](https://mlsanity.com/paper/2504.05295/) (2025)
- [PolarGrad: A Class of Matrix-Gradient Optimizers from a Unifying Preconditioning Perspective](https://mlsanity.com/paper/2505.21799/) (2025)
- [Muon is Not That Special: Random or Inverted Spectra Work Just as Well](https://mlsanity.com/paper/2605.11181/) (2026)
- [Enhancing LLM Training via Spectral Clipping](https://mlsanity.com/paper/2603.14315/) (2026)
- [Why Muon Outperforms Adam: A Curvature Perspective](https://mlsanity.com/paper/2606.04662/) (2026)
- [Softsign: Smooth Sign in Your Optimizer For Better Parameter Heterogeneity Handling](https://mlsanity.com/paper/2605.31371/) (2026)
- [Spectral Scaling Laws of Muon](https://mlsanity.com/paper/2606.04058/) (2026)
- [Dion3: Full-Stack Orthogonal Updates](https://mlsanity.com/paper/2608.11612/) (2026)
- [DynMuon: A Dynamic Spectral Shaping View of Muon](https://mlsanity.com/paper/2605.17109/) (2026)
- [The Spectral Dynamics and Noise Geometry of Muon](https://mlsanity.com/paper/2606.08388/) (2026)
- [Musec: MomentUm SpEctral Clipping for Stable Muon-type Training](https://mlsanity.com/paper/2609.11655/) (2026)
- [Muonp: Muon with Fractional Spectral Powers](https://mlsanity.com/paper/2606.13867/) (2026)
- [Optimizer Memory Schedules for Outscaling the Overtraining Axis](https://mlsanity.com/paper/2609.04577/) (2026)
- [Beyond the Matrix Sign: Quadratic Spectral Descent](https://mlsanity.com/paper/2609.07597/) (2026)

## Abstract (arXiv)

Muon combines current and past gradients into matrix momentum. For M=USigma V^top, the idealized polar update Q=UV^top gives every singular direction the same weight. We refer to this as the flat profile. Several recent optimizers replace this flat profile with fine-grained spectral maps that give each direction its own gain. We ask how much of this spectral detail a Muon update needs. Our spectral diagnostics show that approximately 94--97% of measured singular modes lie below an estimated noise edge, yet collectively align positively with a reference gradient. We introduce BulkBoost, a two-band spectral reweighting framework with fixed-rank and noise-calibrated variants. The latter uses split-minibatch gradient differences to calibrate a Marchenko--Pastur reference edge for Muon's Nesterov input, separating the bulk below the edge from the spikes above it. Both variants increase the bulk's relative weight through one shared gain while preserving the Frobenius norm of each matrix's unreweighted direction. For a fixed partition, our theory gives the first-order condition under which moving weight toward the bulk lowers the loss. It also quantifies the fraction of the maximal first-order improvement rate, over all per-mode reallocations, that two bands can capture. Across 30 continued-pretraining settings spanning Pythia-14M to 410M and six corpora, two-band reweighting is competitive with the fine-grained power-law profile of Freon and outperforms Spectra. Measured against Muon's flat profile, Freon reduces final loss by 0.022% of the pre-adaptation loss on average, whereas the two-band variants achieve reductions of 0.073--0.147%. These observations suggest that useful departures from the flat profile are surprisingly low-dimensional: a single bulk-to-spike gain captures at least as much benefit as the fine-grained spectral profiles.