# Dion: Distributed Orthonormalized Updates

arXiv: https://arxiv.org/abs/2504.05295 · HTML: https://mlsanity.com/paper/2504.05295/

## Key facts

- Cited by 68 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 55 references that are in the map.
- First posted on arXiv 7 Apr 2025.

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

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [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)
- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [Megatron-LM: Training Multi-Billion Parameter Language Models Using Model Parallelism](https://mlsanity.com/paper/1909.08053/) (2019)
- [Root Mean Square Layer Normalization](https://mlsanity.com/paper/1910.07467/) (2019)
- [ZeRO: Memory Optimizations Toward Training Trillion Parameter Models](https://mlsanity.com/paper/1910.02054/) (2019)
- [The FineWeb Datasets: Decanting the Web for the Finest Text Data at Scale](https://mlsanity.com/paper/2406.17557/) (2024)
- [Gemma: Open Models Based on Gemini Research and Technology](https://mlsanity.com/paper/2403.08295/) (2024)
- [PyTorch FSDP: Experiences on Scaling Fully Sharded Data Parallel](https://mlsanity.com/paper/2304.11277/) (2023)
- [DeepSeek LLM: Scaling Open-Source Language Models with Longtermism](https://mlsanity.com/paper/2401.02954/) (2024)
- [Efficient Large-Scale Language Model Training on GPU Clusters Using Megatron-LM](https://mlsanity.com/paper/2104.04473/) (2021)
- [GaLore: Memory-Efficient LLM Training by Gradient Low-Rank Projection](https://mlsanity.com/paper/2403.03507/) (2024)
- [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)
- [Horovod: fast and easy distributed deep learning in TensorFlow](https://mlsanity.com/paper/1802.05799/) (2018)
- [PowerSGD: Practical Low-Rank Gradient Compression for Distributed Optimization](https://mlsanity.com/paper/1905.13727/) (2019)
- [OpenMathInstruct-2: Accelerating AI for Math with Massive Open-Source Instruction Data](https://mlsanity.com/paper/2410.01560/) (2024)
- [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)
- [Primer: Searching for Efficient Transformers for Language Modeling](https://mlsanity.com/paper/2109.08668/) (2021)
- [ATOMO: Communication-efficient Learning via Atomic Sparsification](https://mlsanity.com/paper/1806.04090/) (2018)
- [DiLoCo: Distributed Low-Communication Training of Language Models](https://mlsanity.com/paper/2311.08105/) (2023)
- [A Spectral Condition for Feature Learning](https://mlsanity.com/paper/2310.17813/) (2023)
- [A Distributed Data-Parallel PyTorch Implementation of the Distributed Shampoo Optimizer for Training Neural Networks At-Scale](https://mlsanity.com/paper/2309.06497/) (2023)
- [Modular Duality in Deep Learning](https://mlsanity.com/paper/2410.21265/) (2024)

## Built on by (most cited first, 30 of 68)

- [NorMuon: Making Muon more efficient and scalable](https://mlsanity.com/paper/2510.05491/) (2025)
- [Isotropic Curvature Model for Understanding Deep Learning Optimization: Is Gradient Orthogonalization Optimal?](https://mlsanity.com/paper/2511.00674/) (2025)
- [MuonBP: Faster Muon via Block-Periodic Orthogonalization](https://mlsanity.com/paper/2510.16981/) (2025)
- [Hyperparameter Transfer Enables Consistent Gains of Matrix-Preconditioned Optimizers Across Scales](https://mlsanity.com/paper/2512.05620/) (2025)
- [Low-rank Orthogonalization for Large-scale Matrix Optimization with Applications to Foundation Model Training](https://mlsanity.com/paper/2509.11983/) (2025)
- [Generalized Gradient Norm Clipping & Non-Euclidean (L0,L1)-Smoothness](https://mlsanity.com/paper/2506.01913/) (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)
- [Muon2: Boosting Muon via Adaptive Second-Moment Preconditioning](https://mlsanity.com/paper/2604.09967/) (2026)
- [What Really Matters in Matrix-Whitening Optimizers?](https://mlsanity.com/paper/2510.25000/) (2025)
- [Muon is Not That Special: Random or Inverted Spectra Work Just as Well](https://mlsanity.com/paper/2605.11181/) (2026)
- [Dion2: A Simple Method to Shrink Matrix in Muon](https://mlsanity.com/paper/2512.16928/) (2025)
- [Insights on Muon from Simple Quadratics](https://mlsanity.com/paper/2602.11948/) (2026)
- [Effective Quantization of Muon Optimizer States](https://mlsanity.com/paper/2509.23106/) (2025)
- [Arcee Trinity Large Technical Report](https://mlsanity.com/paper/2602.17004/) (2026)
- [Symmetry-Compatible Principle for Optimizer Design: Embeddings, LM Heads, SwiGLU MLPs, and MoE Routers](https://mlsanity.com/paper/2605.18106/) (2026)
- [FedMuon: Federated Learning with Bias-corrected LMO-based Optimization](https://mlsanity.com/paper/2509.26337/) (2025)
- [Optimal Scaling Needs Optimal Norm](https://mlsanity.com/paper/2510.03871/) (2025)
- [Rethinking Muon Beyond Pretraining: Spectral Failures and High-Pass Remedies for VLA and RLVR](https://mlsanity.com/paper/2605.19282/) (2026)
- [Understanding the Mechanisms of Fast Hyperparameter Transfer](https://mlsanity.com/paper/2512.22768/) (2025)
- [Spherical Leech Quantization for Visual Tokenization and Generation](https://mlsanity.com/paper/2512.14697/) (2025)
- [Why Muon Outperforms Adam: A Curvature Perspective](https://mlsanity.com/paper/2606.04662/) (2026)
- [Optimizer-Model Consistency: Full Finetuning with the Same Optimizer as Pretraining Forgets Less](https://mlsanity.com/paper/2605.06654/) (2026)
- [DeMuon: A Decentralized Muon for Matrix Optimization over Graphs](https://mlsanity.com/paper/2510.01377/) (2025)
- [Ky Fan Norms and Beyond: Dual Norms and Combinations for Matrix Optimization](https://mlsanity.com/paper/2512.09678/) (2025)
- [Learning to Advect: A Neural Semi-Lagrangian Architecture for Weather Forecasting](https://mlsanity.com/paper/2601.21151/) (2026)
- [PRISM: Distribution-free Adaptive Computation of Matrix Functions for Accelerating Neural Network Training](https://mlsanity.com/paper/2601.22137/) (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)
- [Noise-Adaptive Layerwise Learning Rates: Accelerating Geometry-Aware Optimization for Deep Neural Network Training](https://mlsanity.com/paper/2510.14009/) (2025)

## Abstract (arXiv)

Orthonormalized updates accelerate training, improve stability, and enable robust hyperparameter transfer, but existing methods like Muon rely on dense matrix operations that clash with sharded weights in large-scale LLM training, causing high compute and communication cost. We introduce Dion (Distributed Orthonormalization), a scalable and efficient update rule that replaces Newton-Schulz iteration with amortized power iteration on a momentum buffer, avoiding full-matrix reconstruction and integrating cleanly with weight sharding. The rank-fraction parameter with error feedback enables low-rank updates that balance quality with significant cost savings. On language models from 160M to 3B parameters, Dion retains the benefits of orthonormalized updates, while markedly reducing wall-clock time at scale, making it a practical optimizer for next-generation foundation models. Code is available at: https://github.com/microsoft/dion/