# FedMuon: Federated Learning with Bias-corrected LMO-based Optimization

arXiv: https://arxiv.org/abs/2509.26337 · HTML: https://mlsanity.com/paper/2509.26337/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 54 of its 73 references that are in the map.
- First posted on arXiv 30 Sep 2025.

## Builds on (most cited first, 30 of 54)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Advances and Open Problems in Federated Learning](https://mlsanity.com/paper/1912.04977/) (2019)
- [Group Normalization](https://mlsanity.com/paper/1803.08494/) (2018)
- [Adaptive Federated Optimization](https://mlsanity.com/paper/2003.00295/) (2020)
- [Measuring the Effects of Non-Identical Data Distribution for Federated Visual Classification](https://mlsanity.com/paper/1909.06335/) (2019)
- [Local SGD Converges Fast and Communicates Little](https://mlsanity.com/paper/1805.09767/) (2018)
- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [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)
- [Sparsified SGD with Memory](https://mlsanity.com/paper/1809.07599/) (2018)
- [A Unified Theory of Decentralized SGD with Changing Topology and Local Updates](https://mlsanity.com/paper/2003.10422/) (2020)
- [Error Feedback Fixes SignSGD and other Gradient Compression Schemes](https://mlsanity.com/paper/1901.09847/) (2019)
- [The Non-IID Data Quagmire of Decentralized Machine Learning](https://mlsanity.com/paper/1910.00189/) (2019)
- [AdaBelief Optimizer: Adapting Stepsizes by the Belief in Observed Gradients](https://mlsanity.com/paper/2010.07468/) (2020)
- [PowerSGD: Practical Low-Rank Gradient Compression for Distributed Optimization](https://mlsanity.com/paper/1905.13727/) (2019)
- [Training Deep Learning Models with Norm-Constrained LMOs](https://mlsanity.com/paper/2502.07529/) (2025)
- [AdaGrad stepsizes: Sharp convergence over nonconvex landscapes](https://mlsanity.com/paper/1806.01811/) (2018)
- [The Road Less Scheduled](https://mlsanity.com/paper/2405.15682/) (2024)
- [Is Local SGD Better than Minibatch SGD?](https://mlsanity.com/paper/2002.07839/) (2020)
- [Achieving Geometric Convergence for Distributed Optimization over Time-Varying Graphs](https://mlsanity.com/paper/1607.03218/) (2016)
- [D2: Decentralized Training over Decentralized Data](https://mlsanity.com/paper/1803.07068/) (2018)
- [Momentum Improves Normalized SGD](https://mlsanity.com/paper/2002.03305/) (2020)
- [Optimal Client Sampling for Federated Learning](https://mlsanity.com/paper/2010.13723/) (2020)
- [The Polar Express: Optimal Matrix Sign Methods and Their Application to the Muon Algorithm](https://mlsanity.com/paper/2505.16932/) (2025)
- [Understanding Gradient Orthogonalization for Deep Learning via Non-Euclidean Trust-Region Optimization](https://mlsanity.com/paper/2503.12645/) (2025)
- [Communication Compression for Decentralized Training](https://mlsanity.com/paper/1803.06443/) (2018)
- [Sharper Convergence Guarantees for Asynchronous SGD for Distributed and Federated Learning](https://mlsanity.com/paper/2206.08307/) (2022)

## Built on by (most cited first, 8 of 8)

- [DeMuon: A Decentralized Muon for Matrix Optimization over Graphs](https://mlsanity.com/paper/2510.01377/) (2025)
- [CacheMuon: Using Temporal Preconditioning To Approximate Polar Factor](https://mlsanity.com/paper/2606.16371/) (2026)
- [A Note on Stability for Orthogonalized Matrix Momentum with Client Sampling](https://mlsanity.com/paper/2606.01720/) (2026)
- [Sign compression for Muon: SignMuon, MuonSign, and the Limits of Error Feedback](https://mlsanity.com/paper/2607.29674/) (2026)
- [Federated Compositional Muon Optimizer for Matrix-Wise Models](https://mlsanity.com/paper/2608.12710/) (2026)
- [FedSubMuon: Communication-Efficient Federated LLM Fine-Tuning via Structured Subspace Muon](https://mlsanity.com/paper/2609.06073/) (2026)
- [FedMIX-P: Mixing Local and Global Preconditioners for Federated Vision and Language Model Training](https://mlsanity.com/paper/2610.01515/) (2026)
- [Understanding Trajectory Heterogeneity in Federated World Model Learning](https://mlsanity.com/paper/2610.02957/) (2026)

## Abstract (arXiv)

Recently, a new optimization method based on the linear minimization oracle (LMO), called Muon, has been attracting increasing attention since it can train neural networks faster than existing adaptive optimization methods, such as Adam. In this paper, we study how Muon can be utilized in federated learning. We first show that straightforwardly using Muon as the local optimizer of FedAvg does not converge to the stationary point since the LMO is a biased operator. We then propose FedMuon which can mitigate this issue. We also analyze how solving the LMO approximately affects the convergence rate and find that, surprisingly, FedMuon can converge for any number of Newton-Schulz iterations, while it can converge faster as we solve the LMO more accurately. Through experiments, we demonstrated that FedMuon can outperform the state-of-the-art federated learning methods.