# Monarch-Muon: Structure-Aware Muon for Fast and Memory-Efficient Pretraining

arXiv: https://arxiv.org/abs/2610.17429 · HTML: https://mlsanity.com/paper/2610.17429/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 25 references that are in the map.
- First posted on arXiv 12 Oct 2026.

## Placed in

- Mixture-of-experts

## Builds on (most cited first, 19 of 19)

- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [Megatron-LM: Training Multi-Billion Parameter Language Models Using Model Parallelism](https://mlsanity.com/paper/1909.08053/) (2019)
- [The FineWeb Datasets: Decanting the Web for the Finest Text Data at Scale](https://mlsanity.com/paper/2406.17557/) (2024)
- [GaLore: Memory-Efficient LLM Training by Gradient Low-Rank Projection](https://mlsanity.com/paper/2403.03507/) (2024)
- [The Polar Express: Optimal Matrix Sign Methods and Their Application to the Muon Algorithm](https://mlsanity.com/paper/2505.16932/) (2025)
- [Monarch: Expressive Structured Matrices for Efficient and Accurate Training](https://mlsanity.com/paper/2204.00595/) (2022)
- [NorMuon: Making Muon more efficient and scalable](https://mlsanity.com/paper/2510.05491/) (2025)
- [Dion: Distributed Orthonormalized Updates](https://mlsanity.com/paper/2504.05295/) (2025)
- [APOLLO: SGD-like Memory, AdamW-level Performance](https://mlsanity.com/paper/2412.05270/) (2024)
- [Fira: Can We Achieve Full-rank Training of LLMs Under Low-rank Constraint?](https://mlsanity.com/paper/2410.01623/) (2024)
- [MuonBP: Faster Muon via Block-Periodic Orthogonalization](https://mlsanity.com/paper/2510.16981/) (2025)
- [COAT: Compressing Optimizer states and Activation for Memory-Efficient FP8 Training](https://mlsanity.com/paper/2410.19313/) (2024)
- [Compute Better Spent: Replacing Dense Layers with Structured Matrices](https://mlsanity.com/paper/2406.06248/) (2024)
- [FRUGAL: Memory-Efficient Optimization by Reducing State Overhead for Scalable Training](https://mlsanity.com/paper/2411.07837/) (2024)
- [LoRA-Muon: Spectral Steepest Descent on the Low-Rank Manifold](https://mlsanity.com/paper/2606.12921/) (2026)
- [ProcrustesGPT: Compressing LLMs with Structured Matrices and Orthogonal Transformations](https://mlsanity.com/paper/2506.02818/) (2025)
- [DMuon: Efficient Distributed Muon Training with Near-Adam Overhead](https://mlsanity.com/paper/2606.27153/) (2026)

## Abstract (arXiv)

Training large language models is constrained by both wall-clock time and memory footprint, and recent matrix-aware optimizers sharpen this tradeoff: Muon can improve optimization but its dense polar update remains expensive. We propose Monarch-Muon, a structured matrix-aware optimizer for weights parameterized by Monarch factors. The resulting structure reduces weights, gradients, and optimizer state, and exposes small independent update subproblems in place of a single dense orthogonalization; we formalize this through a blockwise trust-region interpretation. In quality runs, Monarch-Muon improves over Monarch-AdamW at fixed structure: 3.06 vs. 3.09 in the dense 257M model and 2.68 vs. 2.77 in the 1B MoE model; the two-block MoE variant also matches dense AdamW while using 15% fewer active and 25.4% fewer total parameters. In 6.89B-parameter systems measurements, Monarch-Muon reduces peak memory from 68.9 to 52.9 GB, Muon optimizer state from 14.60 to 6.77 GB, and the Muon optimizer step from 1288 to 304 ms.