# Convergence of Spectral Descent for Non-smooth Optimization

arXiv: https://arxiv.org/abs/2605.26977 · HTML: https://mlsanity.com/paper/2605.26977/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 66 references that are in the map.
- First posted on arXiv 26 May 2026.

## Builds on (most cited first, 30 of 35)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [DeepSeek-V4: Towards Highly Efficient Million-Token Context Intelligence](https://mlsanity.com/paper/2606.19348/) (2026)
- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [GLM-4.5: Agentic, Reasoning, and Coding (ARC) Foundation Models](https://mlsanity.com/paper/2508.06471/) (2025)
- [Muon is Scalable for LLM Training](https://mlsanity.com/paper/2502.16982/) (2025)
- [Shampoo: Preconditioned Stochastic Tensor Optimization](https://mlsanity.com/paper/1802.09568/) (2018)
- [Training Deep Learning Models with Norm-Constrained LMOs](https://mlsanity.com/paper/2502.07529/) (2025)
- [The Polar Express: Optimal Matrix Sign Methods and Their Application to the Muon Algorithm](https://mlsanity.com/paper/2505.16932/) (2025)
- [Muon Optimizes Under Spectral Norm Constraints](https://mlsanity.com/paper/2506.15054/) (2025)
- [A Note on the Convergence of Muon](https://mlsanity.com/paper/2502.02900/) (2025)
- [Practical Efficiency of Muon for Pretraining](https://mlsanity.com/paper/2505.02222/) (2025)
- [Dion: Distributed Orthonormalized Updates](https://mlsanity.com/paper/2504.05295/) (2025)
- [AdaMuon: Adaptive Muon Optimizer](https://mlsanity.com/paper/2507.11005/) (2025)
- [PolarGrad: A Class of Matrix-Gradient Optimizers from a Unifying Preconditioning Perspective](https://mlsanity.com/paper/2505.21799/) (2025)
- [Lion Secretly Solves Constrained Optimization: As Lyapunov Predicts](https://mlsanity.com/paper/2310.05898/) (2023)
- [Implicit Bias of Spectral Descent and Muon on Multiclass Separable Data](https://mlsanity.com/paper/2502.04664/) (2025)
- [Lions and Muons: Optimization via Stochastic Frank-Wolfe under Heavy-Tailed Noise](https://mlsanity.com/paper/2506.04192/) (2025)
- [Subgradient methods for sharp weakly convex functions](https://mlsanity.com/paper/1803.02461/) (2018)
- [Low-rank matrix recovery with composite optimization: good conditioning and rapid convergence](https://mlsanity.com/paper/1904.10020/) (2019)
- [COSMOS: A Hybrid Adaptive Optimizer for Memory-Efficient Training of LLMs](https://mlsanity.com/paper/2502.17410/) (2025)
- [Preconditioning Benefits of Spectral Orthogonalization in Muon](https://mlsanity.com/paper/2601.13474/) (2026)
- [Solving (most) of a set of quadratic equalities: Composite optimization for robust phase retrieval](https://mlsanity.com/paper/1705.02356/) (2017)
- [AdaGrad Meets Muon: Adaptive Stepsizes for Orthogonal Updates](https://mlsanity.com/paper/2509.02981/) (2025)
- [The nonsmooth landscape of phase retrieval](https://mlsanity.com/paper/1711.03247/) (2017)
- [HTMuon: Improving Muon via Heavy-Tailed Spectral Correction](https://mlsanity.com/paper/2603.10067/) (2026)
- [The Implicit Bias of Adam and Muon on Smooth Homogeneous Neural Networks](https://mlsanity.com/paper/2602.16340/) (2026)
- [Manifold constrained steepest descent for smooth and closed-set optimization](https://mlsanity.com/paper/2601.21487/) (2026)
- [Spectral Gradient Descent Mitigates Anisotropy-Driven Misalignment: A Case Study in Phase Retrieval](https://mlsanity.com/paper/2601.22652/) (2026)
- [A Deterministic Nonsmooth Frank Wolfe Algorithm with Coreset Guarantees](https://mlsanity.com/paper/1708.06714/) (2017)
- [Conda: Column-Normalized Adam for Training Large Language Models Faster](https://mlsanity.com/paper/2509.24218/) (2025)

## Built on by (most cited first, 2 of 2)

- [Musec: MomentUm SpEctral Clipping for Stable Muon-type Training](https://mlsanity.com/paper/2609.11655/) (2026)
- [From Parallel to Sequential: Small-Batch Noise Restores Simplicity Bias of Muon](https://mlsanity.com/paper/2610.16868/) (2026)

## Abstract (arXiv)

The Muon optimizer has recently demonstrated remarkable empirical success in training large language models. However, the theoretical understanding of its mechanisms remains limited. Current convergence guarantees for Muon rely heavily on smoothness assumptions, leaving its non-smooth convergence behavior largely unexplored. In this work, we take a step toward bridging this gap by investigating Spectral Descent (SD), a simplified variant of Muon, together with its truncated counterpart, Truncated Spectral Descent (TSD). Under convexity, Lipschitz continuity, and sharpness conditions, we establish global linear convergence for both SD and TSD in non-smooth convex formulations. We also study regularized variants equipped with decoupled weight decay and derive sublinear convergence guarantees through their connection with Frank-Wolfe methods. Finally, we apply our theoretical framework to robust low-rank matrix recovery under mixed sparse and dense noise regimes and provide rigorous recovery guarantees. Numerical experiments support the theoretical findings and demonstrate the effectiveness of Muon-type methods for non-smooth optimization.