# Dion3: Full-Stack Orthogonal Updates

arXiv: https://arxiv.org/abs/2608.11612 · HTML: https://mlsanity.com/paper/2608.11612/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 33 references that are in the map.
- First posted on arXiv 12 Aug 2026.

## Builds on (most cited first, 14 of 14)

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [PyTorch FSDP: Experiences on Scaling Fully Sharded Data Parallel](https://mlsanity.com/paper/2304.11277/) (2023)
- [GLM-5: from Vibe Coding to Agentic Engineering](https://mlsanity.com/paper/2602.15763/) (2026)
- [Muon is Scalable for LLM Training](https://mlsanity.com/paper/2502.16982/) (2025)
- [The Polar Express: Optimal Matrix Sign Methods and Their Application to the Muon Algorithm](https://mlsanity.com/paper/2505.16932/) (2025)
- [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)
- [Nemotron-CLIMB: CLustering-based Iterative Data Mixture Bootstrapping for Language Model Pre-training](https://mlsanity.com/paper/2504.13161/) (2025)
- [MuonBP: Faster Muon via Block-Periodic Orthogonalization](https://mlsanity.com/paper/2510.16981/) (2025)
- [SonicMoE: Accelerating MoE with IO and Tile-aware Optimizations](https://mlsanity.com/paper/2512.14080/) (2025)
- [On Surprising Effectiveness of Masking Updates in Adaptive Optimizers](https://mlsanity.com/paper/2602.15322/) (2026)
- [Motif 2 12.7B technical report](https://mlsanity.com/paper/2511.07464/) (2025)
- [Hierarchical Muon: Tiled Newton-Schulz Updates for Efficient Muon Optimization](https://mlsanity.com/paper/2606.27216/) (2026)
- [Laguna M.1/XS.2 Technical Report](https://mlsanity.com/paper/2605.27605/) (2026)

## Built on by (most cited first, 4 of 4)

- [Nonsmooth Optimization via Orthogonalized Momentum](https://mlsanity.com/paper/2609.13677/) (2026)
- [Cost-free Spectral Estimation for Adaptive Newton--Schulz in Matrix Optimizers](https://mlsanity.com/paper/2609.33047/) (2026)
- [The Row Normalization Puzzle in Muon](https://mlsanity.com/paper/2609.39114/) (2026)
- [Does Muon Need Fine-Grained Spectral Shaping?](https://mlsanity.com/paper/2610.07497/) (2026)

## Abstract (arXiv)

The Muon optimizer incurs a significant overhead cost due to its cubic-time Newton-Schulz orthogonalization step. When weights are sharded, communication overhead compounds this computational cost, eroding the benefits of Muon in many settings. We present Dion3, a revision of Muon that targets this overhead at every level of the stack. Our Gram Newton-Schulz algorithm reduces the FLOP cost of orthogonalization, our CuteDSL kernels accelerate it by exploiting symmetry, and our megabatching strategy reduces communication overhead. Moreover, we propose a simple change to the update rule that cuts costs even further: selecting only a fraction of the momentum matrix's rows to orthogonalize at each step. This update rule improves on Dion (another "compressed" version of Muon), in both speed and performance. Overall, Dion3 matches or improves on the loss achieved by Muon but reduces optimizer step time by up to 6x. Dion3 is available via the dion package (https://github.com/microsoft/dion) as a drop-in replacement for Muon.