arXiv:2605.26977v1 Announce Type: new Abstract: The Muon optimizer has recently demonstrated remarkable empirical success in training large language models. However, the theoretical understanding of its mechanisms remains limited. Current convergence guarantees for Muon rely heavily on smoothness assumptions, leaving its non-smooth convergence behavior largely unexplored. In this work, we take a step toward bridging this gap by investigating Spectral Descent (SD), a simplified variant of Muon, together with its truncated counterpart, Truncated Spectral Descent (TSD). Under convexity, Lipschitz
The rapid ascent of large language models (LLMs) has outpaced theoretical understanding of their underlying optimization methods, creating an urgent need for validated convergence guarantees for novel optimizers like Muon.
Improved theoretical understanding of emergent AI optimization techniques can accelerate model development, reduce computational costs, and allow for more predictable and stable training of advanced AI systems.
The theoretical foundation for optimizing large language models is incrementally strengthened, potentially paving the way for more robust and efficient future AI architectures.
- · AI researchers
- · Large language model developers
- · Computational statisticians
- · Developers of less efficient optimization methods
This research provides a step toward establishing convergence guarantees for Muon, a successful optimizer for LLMs.
Improved theoretical understanding and convergence guarantees could lead to faster and more stable development cycles for next-generation AI models.
More efficient and reliable AI training processes might accelerate progress towards advanced AI capabilities and agentic systems.
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Read at arXiv cs.LG