# LiMuon: Light and Fast Muon Optimizer for Large Models

arXiv: https://arxiv.org/abs/2509.14562 · HTML: https://mlsanity.com/paper/2509.14562/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 40 references that are in the map.
- First posted on arXiv 18 Sep 2025.

## Builds on (most cited first, 27 of 27)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [DistilBERT, a distilled version of BERT: smaller, faster, cheaper and lighter](https://mlsanity.com/paper/1910.01108/) (2019)
- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [Symbolic Discovery of Optimization Algorithms](https://mlsanity.com/paper/2302.06675/) (2023)
- [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)
- [Why gradient clipping accelerates training: A theoretical justification for adaptivity](https://mlsanity.com/paper/1905.11881/) (2019)
- [Momentum-Based Variance Reduction in Non-Convex SGD](https://mlsanity.com/paper/1905.10018/) (2019)
- [Old Optimizer, New Norm: An Anthology](https://mlsanity.com/paper/2409.20325/) (2024)
- [Training Deep Learning Models with Norm-Constrained LMOs](https://mlsanity.com/paper/2502.07529/) (2025)
- [Understanding Gradient Orthogonalization for Deep Learning via Non-Euclidean Trust-Region Optimization](https://mlsanity.com/paper/2503.12645/) (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)
- [The MiniPile Challenge for Data-Efficient Language Models](https://mlsanity.com/paper/2304.08442/) (2023)
- [Gluon: Making Muon & Scion Great Again! (Bridging Theory and Practice of LMO-based Optimizers for LLMs)](https://mlsanity.com/paper/2505.13416/) (2025)
- [Neuro-Symbolic Language Modeling with Automaton-augmented Retrieval](https://mlsanity.com/paper/2201.12431/) (2022)
- [ASGO: Adaptive Structured Gradient Optimization](https://mlsanity.com/paper/2503.20762/) (2025)
- [PolarGrad: A Class of Matrix-Gradient Optimizers from a Unifying Preconditioning Perspective](https://mlsanity.com/paper/2505.21799/) (2025)
- [Lions and Muons: Optimization via Stochastic Frank-Wolfe under Heavy-Tailed Noise](https://mlsanity.com/paper/2506.04192/) (2025)
- [Variance-Reduced and Projection-Free Stochastic Optimization](https://mlsanity.com/paper/1602.02101/) (2016)
- [A Hybrid Stochastic Optimization Framework for Stochastic Composite Nonconvex Optimization](https://mlsanity.com/paper/1907.03793/) (2019)
- [Orthogonalising gradients to speed up neural network optimisation](https://mlsanity.com/paper/2202.07052/) (2022)
- [SUMO: Subspace-Aware Moment-Orthogonalization for Accelerating Memory-Efficient LLM Training](https://mlsanity.com/paper/2505.24749/) (2025)
- [Generalized Gradient Norm Clipping & Non-Euclidean (L0,L1)-Smoothness](https://mlsanity.com/paper/2506.01913/) (2025)

## Built on by (most cited first, 22 of 22)

- [Isotropic Curvature Model for Understanding Deep Learning Optimization: Is Gradient Orthogonalization Optimal?](https://mlsanity.com/paper/2511.00674/) (2025)
- [On the Convergence of Muon and Beyond](https://mlsanity.com/paper/2509.15816/) (2025)
- [Mousse: Rectifying the Geometry of Muon with Curvature-Aware Preconditioning](https://mlsanity.com/paper/2603.09697/) (2026)
- [MuonEq: Balancing Before Orthogonalization with Lightweight Equilibration](https://mlsanity.com/paper/2603.28254/) (2026)
- [ROOT: Robust Orthogonalized Optimizer for Neural Network Training](https://mlsanity.com/paper/2511.20626/) (2025)
- [Improved Convergence Rates of Muon Optimizer for Nonconvex Optimization](https://mlsanity.com/paper/2601.19400/) (2026)
- [Non-Euclidean SGD for Structured Optimization: Unified Analysis and Improved Rates](https://mlsanity.com/paper/2511.11466/) (2025)
- [Muon is Provably Faster with Momentum Variance Reduction](https://mlsanity.com/paper/2512.16598/) (2025)
- [Where Does Warm-Up Come From? Adaptive Scheduling for Norm-Constrained Optimizers](https://mlsanity.com/paper/2602.05813/) (2026)
- [PRISM: Distribution-free Adaptive Computation of Matrix Functions for Accelerating Neural Network Training](https://mlsanity.com/paper/2601.22137/) (2026)
- [Softsign: Smooth Sign in Your Optimizer For Better Parameter Heterogeneity Handling](https://mlsanity.com/paper/2605.31371/) (2026)
- [Communication-Efficient Gluon in Federated Learning](https://mlsanity.com/paper/2604.10689/) (2026)
- [MiMuon: Mixed Muon Optimizer with Improved Generalization for Large Models](https://mlsanity.com/paper/2605.19619/) (2026)
- [When and Why SignSGD Outperforms SGD: A Theoretical Study Based on ℓ1-norm Lower Bounds](https://mlsanity.com/paper/2605.06615/) (2026)
- [Tight Complexity of Muon under Heavy-Tailed Noise](https://mlsanity.com/paper/2605.18528/) (2026)
- [LionMuon: Alternating Spectral and Sign Descent for Efficient Training](https://mlsanity.com/paper/2605.19811/) (2026)
- [OptMuon: Closed-Loop Orthogonalized Momentum Methods for Stochastic Optimization with Zero-Noise Optimality](https://mlsanity.com/paper/2606.08783/) (2026)
- [Spectral Allocation: Why Muon Outperforms Adam, and How to Improve Muon](https://mlsanity.com/paper/2608.25990/) (2026)
- [Zero-order Parameter-free Optimization for LMO-based Methods: Novel Approach for Efficient Fine-tuning](https://mlsanity.com/paper/2606.14970/) (2026)
- [Federated Compositional Muon Optimizer for Matrix-Wise Models](https://mlsanity.com/paper/2608.12710/) (2026)
- [LionMuon: Alternating Spectral and Sign Descent for Efficient Training](https://mlsanity.com/paper/2609.35297/) (2026)
- [Memory-Efficient Subspace Optimization with Residual Correction: From Momentum SGD to Muon](https://mlsanity.com/paper/2610.19844/) (2026)

## Abstract (arXiv)

Large models recently are widely applied in machine learning, so efficient training of large models has received widespread attention. More recently, the useful Muon optimizer is specifically designed for matrix-structured parameters of large models. Although some works have begun to study the Muon optimizer, the existing Muon and its variants still suffer from high sample complexity or high memory for large models. To fill this gap, we propose a light and fast Muon (LiMuon) optimizer for training large models, which builds on the momentum-based variance reduced technique and randomized Singular Value Decomposition (SVD). In particular, our LiMuon simultaneously has a lower memory and lower sample complexity than the Muon and its variants. Moreover, we prove that our LiMuon with lower memory has a lower sample complexity of O(ε-3) for finding an ε-stationary solution of non-convex stochastic optimization under the generalized smoothness condition. To further narrow practice and theory gap, we also prove that our LiMuon with Newton-Schulz steps has a lower sample complexity than the Muon with Newton-Schulz steps. Numerical experimental results on pre-training Mamba-130M, Qwen2.5-0.5B and ViT models demonstrate effectiveness of our LiMuon.