Muon Optimizes Under Spectral Norm Constraints (2506.15054v1)

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.

• The paper establishes that Muon implicitly enforces spectral norm constraints through its integration into the Lion-K framework.
• The analysis demonstrates that Muon, with decoupled weight decay, converges to KKT points in both deterministic and stochastic settings.
• By extending the optimizer via convex mappings, the study unveils new avenues for developing tailored regularization strategies in large-scale model training.

Overview of "Muon Optimizes Under Spectral Norm Constraints"

The paper "Muon Optimizes Under Spectral Norm Constraints" presents an in-depth theoretical analysis of the Muon optimizer, building upon recent empirical successes in optimization algorithms within deep learning frameworks. The authors integrate Muon into the broader Lion-$\mathcal{K}$ framework and reveal the underlying theoretical foundations which explain Muon's implicit regularization effects. This paper is pivotal in understanding Muon's behavior under spectral norm constraints, facilitating its potential applications and generalizations in large-scale model training.

Key Contributions

1. Theoretical Connection to Lion-$\mathcal{K}$: The paper clarifies that Muon can be viewed as belonging to the Lion-$\mathcal{K}$ class, specifically utilizing the nuclear norm. This insight is critical as it positions Muon within a well-studied optimization framework, allowing leveraging of existing theoretical results to explain Muon's performance characteristics.
2. Implicit Constrained Optimization: By applying the Lion-$\mathcal{K}$ theoretical foundation, the authors ascertain that Muon, with decoupled weight decay, implicitly solves an optimization problem enforcing spectral norm constraints on weight matrices. This implicit constraint strengthens Muon's capability to regularize models beyond traditional methods, offering new avenues for optimal weight management during training.
3. Convergence Analysis: The paper delivers a rigorous convergence analysis, discussing convergence rates for Muon both in deterministic and stochastic gradient settings. The establishment of convergence to Karush--Kuhn--Tucker (KKT) points for the constrained optimization problem marked a significant development in optimizing model constraints through adaptive iterations.
4. Generalizations via Convex Maps: By situating Muon within the Lion framework, the paper naturally extends its capacity to manipulate various convex mappings, thereby generalizing the optimizer into a robust family of potentially tailor-made adaptations for distinct training challenges or model-specific regularization needs.

Technical Implications

This paper addresses essential theoretical questions surrounding Muon, contributing significantly toward explaining its efficiency and reliability in empirical studies. The findings suggest that Muon implicitly conducts spectral regularization, promoting particular eigenvalue constraints which are beneficial in minimizing overfitting and enhancing generalization within deep learning models.

Moreover, these results imply strategic advancements in optimizer design, encouraging further exploration of advanced spectral constraint mechanisms and adaptive learning strategies within AI applications. Future endeavors could focus on verifying these results in more diverse training settings and expanding the framework to other complex learning paradigms.

Conclusion

The authors of this paper provide a solid theoretical foundation to understand the Muon's behavior, presenting clear evidence for its implicit spectral norm constraints within the Lion-$\mathcal{K}$ framework. While the practical implications remain to be fully explored across varied models and settings, the established convergence guarantees and potential for generalization pave the way for future developments in adaptive optimization strategies in AI. The transparent integration of Muon into a broader theoretical context ensures its place in continuing improvements in optimization processes that are critical for large-scale model training.