How the Optimizer Shapes Learned Solutions in Equivariant Neural Networks

Researchers demonstrate that the Muon optimizer significantly outperforms Adam when training equivariant neural networks, which encode geometric symmetries by design. Analysis of trained models reveals Muon produces solutions with more regular loss surfaces, higher weight ranks, and better-conditioned representations, suggesting optimizer choice substantially influences how neural networks learn geometric constraints.

Equivariant neural networks represent an important research direction for incorporating geometric structure directly into model architecture, particularly valuable for point cloud and molecular data where symmetries are mathematically fundamental. However, practitioners have consistently observed these constrained architectures are harder to optimize and sometimes underperform less structured alternatives, creating a practical bottleneck despite their theoretical elegance. This work shifts focus from architectural modifications toward the often-overlooked role of optimization algorithms in shaping how equivariant models learn and generalize.

The empirical comparison between Muon and Adam across multiple equivariant architectures on ModelNet40 and molecular datasets provides concrete evidence that optimizer selection matters substantially for this problem class. Rather than merely reporting performance metrics, the authors conduct deeper analysis through Hessian curvature estimation and loss surface visualization, revealing that Muon's solutions inhabit meaningfully different regions of the parameter space—ones with higher-rank learned weights and more regular geometric structure in the loss landscape. These mechanistic insights suggest the interaction between optimization dynamics and geometric inductive bias deserves systematic study.

For the machine learning community, this finding has practical implications: practitioners deploying equivariant architectures might achieve substantial improvements simply by switching optimizers, potentially without architectural redesign. The result also highlights an underexplored avenue for improving constrained neural network training broadly. Going forward, the field should investigate whether Muon's advantages stem from its handling of weight structure, curvature properties, or other factors, and whether similar principles apply to other constraint-respecting architectures like normalizing flows or graph neural networks.

• →Muon optimizer consistently outperforms Adam across equivariant neural network architectures on geometric learning tasks.
• →Solutions found by Muon exhibit more regular loss surfaces and higher-rank learned representations despite larger Hessian curvature.
• →Optimizer choice significantly influences how neural networks satisfy geometric constraints during training.
• →Architectural modifications for equivariant networks may be less critical than previously assumed if paired with appropriate optimizers.
• →The interaction between optimization algorithms and geometric inductive bias remains substantially underexplored in the research literature.