Dimension-Free Saddle-Point Escape in Muon

TL;DR

This paper demonstrates that the Muon optimizer can escape high-dimensional saddle points in large language model training without the typical dimensionality limitations, using advanced spectral analysis techniques.

Contribution

The paper introduces a theoretical framework showing Muon's ability to bypass the curse of dimensionality in saddle-point escape through a novel spectral shaping mechanism.

Findings

Muon achieves dimension-free saddle-point escape in high-dimensional landscapes.

Theoretical analysis proves Muon's escape mechanism is robust against the curse of dimensionality.

Muon's escape bound is algebraically independent of the problem dimension.

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.