Dimension-Free Saddle-Point Escape in Muon

2026-05-10

AI Summary

This work addresses the convergence bottleneck in large language model training caused by pathological flat saddle points within high-dimensional non-convex optimization landscapes. It establishes, for the first time, a dimension-independent saddle-point escape theory for the Muon optimizer. By leveraging generalized matrix perturbation theory, resolvent functional calculus, and macroscopic Cauchy contour integration—while dispensing with isotropic noise assumptions and Tracy–Widom edge singularities—the study reveals that Muon achieves nonequilibrium trajectory evolution through a nonlinear spectral shaping mechanism. Under a sufficient spectral gap condition, the analysis proves that Muon attains deterministic O(1) discrete ballistic ejection, yielding an algebraically derived, dimension-independent upper bound on escape time. This result overcomes the dimensional scalability limitations inherent to conventional adaptive optimizers and ensures trajectory stability via structural incoherence.

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.

Problem

• saddle-point escape
• high-dimensional optimization
• large language models
• dimensional curse
• non-convex optimization

Innovation

• dimension-free optimization
• saddle-point escape
• Muon optimizer
• nonlinear spectral shaping
• matrix perturbation theory