# PolarGrad: A Class of Matrix-Gradient Optimizers from a Unifying Preconditioning Perspective

arXiv: https://arxiv.org/abs/2505.21799 · HTML: https://mlsanity.com/paper/2505.21799/

## Key facts

- Cited by 52 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 May 2025.

## Built on by (most cited first, 30 of 52)

- [Muon Outperforms Adam in Tail-End Associative Memory Learning](https://mlsanity.com/paper/2509.26030/) (2025)
- [Lions and Muons: Optimization via Stochastic Frank-Wolfe under Heavy-Tailed Noise](https://mlsanity.com/paper/2506.04192/) (2025)
- [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)
- [Preconditioning Benefits of Spectral Orthogonalization in Muon](https://mlsanity.com/paper/2601.13474/) (2026)
- [An Exploration of Non-Euclidean Gradient Descent: Muon and its Many Variants](https://mlsanity.com/paper/2510.09827/) (2025)
- [AdaGrad Meets Muon: Adaptive Stepsizes for Orthogonal Updates](https://mlsanity.com/paper/2509.02981/) (2025)
- [LiMuon: Light and Fast Muon Optimizer for Large Models](https://mlsanity.com/paper/2509.14562/) (2025)
- [Low-rank Orthogonalization for Large-scale Matrix Optimization with Applications to Foundation Model Training](https://mlsanity.com/paper/2509.11983/) (2025)
- [The Newton-Muon Optimizer](https://mlsanity.com/paper/2604.01472/) (2026)
- [Mousse: Rectifying the Geometry of Muon with Curvature-Aware Preconditioning](https://mlsanity.com/paper/2603.09697/) (2026)
- [What Really Matters in Matrix-Whitening Optimizers?](https://mlsanity.com/paper/2510.25000/) (2025)
- [Sharp Capacity Scaling of Spectral Optimizers in Learning Associative Memory](https://mlsanity.com/paper/2603.26554/) (2026)
- [Cautious Weight Decay](https://mlsanity.com/paper/2510.12402/) (2025)
- [Insights on Muon from Simple Quadratics](https://mlsanity.com/paper/2602.11948/) (2026)
- [FISMO: Fisher-Structured Momentum-Orthogonalized Optimizer](https://mlsanity.com/paper/2601.21750/) (2026)
- [Symmetry-Compatible Principle for Optimizer Design: Embeddings, LM Heads, SwiGLU MLPs, and MoE Routers](https://mlsanity.com/paper/2605.18106/) (2026)
- [MARS-M: When Variance Reduction Meets Matrices](https://mlsanity.com/paper/2510.21800/) (2025)
- [Delving into Muon and Beyond: Deep Analysis and Extensions](https://mlsanity.com/paper/2602.04669/) (2026)
- [Muon Does Not Converge on Convex Lipschitz Functions](https://mlsanity.com/paper/2605.08980/) (2026)
- [Non-Euclidean SGD for Structured Optimization: Unified Analysis and Improved Rates](https://mlsanity.com/paper/2511.11466/) (2025)
- [Accelerating LLM Pre-Training through Flat-Direction Dynamics Enhancement](https://mlsanity.com/paper/2602.22681/) (2026)
- [Spectral Gradient Descent Mitigates Anisotropy-Driven Misalignment: A Case Study in Phase Retrieval](https://mlsanity.com/paper/2601.22652/) (2026)
- [Why Muon Outperforms Adam: A Curvature Perspective](https://mlsanity.com/paper/2606.04662/) (2026)
- [A Tale of Two Geometries: Adaptive Optimizers and Non-Euclidean Descent](https://mlsanity.com/paper/2511.20584/) (2025)
- [DeMuon: A Decentralized Muon for Matrix Optimization over Graphs](https://mlsanity.com/paper/2510.01377/) (2025)
- [Spectra: Rethinking Optimizers for LLMs Under Spectral Anisotropy](https://mlsanity.com/paper/2602.11185/) (2026)
- [Beyond Muon: MUD (MomentUm Decorrelation) for Faster Transformer Training](https://mlsanity.com/paper/2603.17970/) (2026)
- [DynMuon: A Dynamic Spectral Shaping View of Muon](https://mlsanity.com/paper/2605.17109/) (2026)
- [The Spectral Dynamics and Noise Geometry of Muon](https://mlsanity.com/paper/2606.08388/) (2026)

## Abstract (arXiv)

The ever-growing scale of deep learning models and training data underscores the critical importance of efficient optimization methods. While preconditioned gradient methods such as Adam and AdamW are the de facto optimizers for training neural networks and large language models, structure-aware preconditioned optimizers like Shampoo and Muon, which utilize the matrix structure of gradients, have demonstrated promising evidence of faster convergence. In this paper, we introduce a unifying framework for analyzing "matrix-aware" preconditioned methods, which not only sheds light on the effectiveness of Muon and related optimizers but also leads to a class of new structure-aware preconditioned methods. A key contribution of this framework is its precise distinction between preconditioning strategies that treat neural network weights as vectors (addressing curvature anisotropy) versus those that consider their matrix structure (addressing gradient anisotropy). This perspective provides new insights into several empirical phenomena in language model pre-training, including Adam's training instabilities, Muon's accelerated convergence, and the necessity of learning rate warmup for Adam. Building upon this framework, we introduce PolarGrad, a new class of preconditioned optimization methods based on the polar decomposition of matrix-valued gradients. As a special instance, PolarGrad includes Muon with updates scaled by the nuclear norm of the gradients. We provide numerical implementations of these methods, leveraging efficient numerical polar decomposition algorithms for enhanced convergence. Our extensive evaluations across diverse matrix optimization problems and language model pre-training tasks demonstrate that PolarGrad outperforms both Adam and Muon.