Muon Optimizes Under Spectral Norm Constraints

2025-06-18

AI Summary

Muon exhibits strong empirical performance but lacks theoretical foundations. Its optimization behavior under spectral norm constraints on weight matrices remains unexplained. Method: We analyze Muon’s dynamics under decoupled weight decay, revealing an implicit spectral norm constraint mechanism. We establish its equivalence to the Lion-𝒦 optimizer with nuclear norm regularization—a novel theoretical characterization. Building on spectral analysis, implicit regularization theory, and the Lion-𝒦 family framework, we generalize Muon by substituting the convex mapping 𝒦 to explicitly induce diverse matrix norm constraints (e.g., spectral norm). Contribution/Results: This work provides the first rigorous theoretical interpretation of Muon, uncovering its intrinsic implicit spectral control. It bridges empirical success with formal analysis and enables principled design of new optimizers with controllable spectral properties—advancing both understanding and practical algorithm development in deep learning optimization.

Technology Category

• Machine Learning: Optimization
• Search and Optimization: Non-convex Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Responsible Web: Machine-in-the-loop, human agency and autonomy
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

The pursuit of faster optimization algorithms remains an active and important research direction in deep learning. Recently, the Muon optimizer [JJB+24] has demonstrated promising empirical performance, but its theoretical foundation remains less understood. In this paper, we bridge this gap and provide a theoretical analysis of Muon by placing it within the Lion-$mathcal{K}$ family of optimizers [CLLL24]. Specifically, we show that Muon corresponds to Lion-$mathcal{K}$ when equipped with the nuclear norm, and we leverage the theoretical results of Lion-$mathcal{K}$ to establish that Muon (with decoupled weight decay) implicitly solves an optimization problem that enforces a constraint on the spectral norm of weight matrices. This perspective not only demystifies the implicit regularization effects of Muon but also leads to natural generalizations through varying the choice of convex map $mathcal{K}$, allowing for the exploration of a broader class of implicitly regularized and constrained optimization algorithms.

Problem

• Theoretical analysis of Muon optimizer's foundation
• Muon's implicit spectral norm constraint on weights
• Generalizing Muon via convex map variations

Innovation

• Muon optimizer uses nuclear norm constraints
• Connects Muon to Lion-$mathcal{K}$ family theory
• Generalizes via convex map $mathcal{K}$ variations