Finite Newton-Schulz Improves LLM Optimizer Convergence

Mingyi Li, Taira Tsuchiya· August 28, 2026 View original

Key takeaways

  • Finite Newton-Schulz iterations in Muon optimizer can be beneficial for nonsmooth nonconvex optimization.
  • The finite depth acts as a smoothing mechanism for the polar map, aiding convergence.
  • This approach can lead to guaranteed convergence to stationary points with optimal sample complexity.
  • The findings offer insights for improving LLM training stability and efficiency.

Who benefits

AI/ML DevelopmentCloud ComputingResearch & DevelopmentHigh-Performance Computing

Summary

This research shows that using a finite number of Newton-Schulz iterations in the Muon optimizer, which is crucial for large language model pretraining, can actually benefit nonsmooth nonconvex optimization. It proves that this finite depth smooths the polar map, leading to guaranteed convergence to stationary points with optimal sample complexity.

The Muon optimizer is widely used for training large language models, particularly for matrix-valued parameters, by approximately orthogonalizing its momentum using a few Newton-Schulz iterations. Previous theoretical understanding often treated this finite iteration depth as an approximation error, suggesting it might hinder performance. However, new research demonstrates that this finite Newton-Schulz iteration can be advantageous, especially in nonsmooth nonconvex optimization. It effectively smooths the discontinuous polar map into a Lipschitz map, which is beneficial for convergence. The study proves that a logarithmically growing Newton-Schulz depth is sufficient for convergence to stationary points, matching optimal bounds for such optimization problems.

Why it matters

This finding provides a deeper theoretical understanding of a key optimization technique used in LLM training, potentially guiding future improvements in model stability and training efficiency, especially for complex, nonsmooth objectives.

How to implement this in your domain

  1. 1Review current LLM training pipelines to understand the specific optimizers and their configurations.
  2. 2Experiment with varying Newton-Schulz iteration depths in Muon-like optimizers for custom LLM pretraining tasks.
  3. 3Analyze the impact of different smoothing properties on convergence rates and model performance for nonsmooth objectives.
  4. 4Consider this theoretical insight when debugging or optimizing training runs that encounter convergence issues.

Original post by Mingyi Li, Taira Tsuchiya

"arXiv:2608.26288v1 Announce Type: new Abstract: Muon has emerged as a strong optimizer for the matrix-valued parameters in large language model pretraining, approximately orthogonalizing its momentum with a few Newton-Schulz iterations. Existing theory either replaces this iterat…"

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Originally posted by Mingyi Li, Taira Tsuchiya on X · view source

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