Cost-free Spectral Estimation for Adaptive Newton--Schulz in Matrix Optimizers

arXiv:2609.33047 · cs.LG · Submitted 2026-09-27 · Read on arXiv

cs.LG

Submitted: 2026-09-27

Updated: 2026-09-27

Code: https://github.com/tilde-research/online-kl-shampoo-release

Terminology

• Accuracy — 12× in this paper · explained in 1 episode
• Polar Express — 7× in this paper
• Regret — 6× in this paper · explained in 3 episodes
• Trace — 5× in this paper · explained in 2 episodes
• Adaptation — 3× in this paper · explained in 1 episode
• Embeddings — 2× in this paper · explained in 2 episodes
• Ablation — 1× in this paper · explained in 1 episode
• Adam — 1× in this paper · explained in 2 episodes
• Benchmark — 1× in this paper · explained in 2 episodes
• Regularization — 1× in this paper · explained in 3 episodes
• Transformer — 1× in this paper · explained in 2 episodes

Sources

• Dion: Distributed Orthonormalized Updates
• The Polar Express: Optimal Matrix Sign Methods and Their Application to the Muon Algorithm
• Dion3: Full-Stack Orthogonal Updates
• Scalable Second Order Optimization for Deep Learning
• Sublinear Time Spectral Density Estimation
• Muon with Nesterov Momentum: Heavy-Tailed Noise and (Randomized) Inexact Polar Decomposition
• DeepSeek-V4: Towards Highly Efficient Million-Token Context Intelligence
• CacheMuon: Using Temporal Preconditioning To Approximate Polar Factor
• On MUON optimization: From non-convergence to an error analysis with Polar Express and the Newton-Schulz polynomial from implementations
• Practical Efficiency of Muon for Pretraining
• GLM-5: from Vibe Coding to Agentic Engineering
• Accelerating Newton-Schulz Iteration for Orthogonalization via Chebyshev-type Polynomials
• ROOT: Robust Orthogonalized Optimizer for Neural Network Training
• MuonBP: Faster Muon via Block-Periodic Orthogonalization
• Convergence of Muon with Newton-Schulz
• Kimi K3: Open Frontier Intelligence
• Muon is Scalable for LLM Training
• Decoupled Weight Decay Regularization
• NVIDIA Nemotron Nano 2: An Accurate and Efficient Hybrid Mamba-Transformer Reasoning Model
• The FineWeb Datasets: Decanting the Web for the Finest Text Data at Scale

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