Dimension-Free Saddle-Point Escape in Muon

• URL: http://arxiv.org/abs/2605.09331v1
• Date: Sun, 10 May 2026 05:06:07 GMT
• Title: Dimension-Free Saddle-Point Escape in Muon
• Abstract summary: Modern Large Language Model (LLM) training is bottlenecked by pathologically flat saddle points in extreme high-dimensional landscapes.<n>Motivated by this challenge, we analyze the saddle-point escape dynamics of the emerging Muon.
• Score: 10.480246288279517
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Modern Large Language Model (LLM) training is fundamentally bottlenecked by pathologically flat saddle points in extreme high-dimensional landscapes. Motivated by this challenge, we analyze the saddle-point escape dynamics of the emerging Muon optimizer, demonstrating its resilience against the $\mathcal{O}(D)$ dimensional curse that severely traps element-wise adaptive optimizers like AdamW. By extending generalized matrix perturbation theory, we develop a theoretical framework to capture Muon's non-equilibrium optimization trajectories. This theoretical machinery mathematically proves that Muon elegantly bypasses the dimensional curse via a non-linear spectral shaping mechanism. By leveraging resolvent functional calculus and macroscopic Cauchy contour integration, we avoid isotropic noise assumptions and Tracy-Widom edge singularities. We establish that structural incoherence securely shields the trajectory from orthogonal drift, enabling a dimension-free saddle-point escape, and triggering a deterministic $\mathcal{O}(1)$ discrete ballistic ejection under sufficient spectral gap. Consequently, we provide an algebraically dimension-free escape bound for Muon, formalizing the underlying mechanics of its non-convex optimization dynamics.

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