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Dimension-Free Saddle-Point Escape in Muon

arXiv 2605.09331 ↗First posted 10 May 2026cs.LGcited by 0

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Abstract (arXiv)

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 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 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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References · 23

  1. 2020Scaling Laws for Neural Language Modelscited by 6,411
  2. 2018Shampoo: Preconditioned Stochastic Tensor Optimizationcited by 413
  3. 2021Attention is Not All You Need: Pure Attention Loses Rank Doubly Exponentially with Depthcited by 411
  4. 2014Identifying and attacking the saddle point problem in high-dimensional non-convex optimizationcited by 286
  5. 2017How to Escape Saddle Points Efficientlycited by 284
  6. 2014The Loss Surfaces of Multilayer Networkscited by 278
  7. 2015Escaping From Saddle Points --- Online Stochastic Gradient for Tensor Decompositioncited by 241
  8. 2025Understanding Gradient Orthogonalization for Deep Learning via Non-Euclidean Trust-Region Optimizationcited by 81
  9. 2018Escaping Saddles with Stochastic Gradientscited by 66
  10. 2018Stochastic Nested Variance Reduction for Nonconvex Optimizationcited by 63
  11. 2020Adaptive Inertia: Disentangling the Effects of Adaptive Learning Rate and Momentumcited by 40
  12. 2025Muon Outperforms Adam in Tail-End Associative Memory Learningcited by 38
11 more
  1. 2018Large-Scale Distributed Second-Order Optimization Using Kronecker-Factored Approximate Curvature for Deep Convolutional Neural Networkscited by 37
  2. 2019Escaping Saddle Points with Adaptive Gradient Methodscited by 34
  3. 2021Positive-Negative Momentum: Manipulating Stochastic Gradient Noise to Improve Generalizationcited by 28
  4. 2025On the Convergence of Muon and Beyondcited by 28
  5. 2026Mousse: Rectifying the Geometry of Muon with Curvature-Aware Preconditioningcited by 15
  6. 2017A Generic Approach for Escaping Saddle pointscited by 13
  7. 2022On the Overlooked Structure of Stochastic Gradientscited by 13
  8. 2025ROOT: Robust Orthogonalized Optimizer for Neural Network Trainingcited by 13
  9. 2017Behavior of Accelerated Gradient Methods Near Critical Points of Nonconvex Functionscited by 7
  10. 2026PRISM: Structured Optimization via Anisotropic Spectral Shapingcited by 2
  11. 2024Analysis of singular subspaces under random perturbationscited by 1

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