Ky Fan Norms and Beyond: Dual Norms and Combinations for Matrix Optimization

Researchers introduce the Fanion family of optimization algorithms that extend beyond spectral norms used in the Muon optimizer, leveraging Ky Fan norm duals for matrix optimization in deep learning. Two variants, F-Muon and S-Muon, match or exceed Muon's performance across diverse tasks, with particular improvements on synthetic convex problems.

This research addresses a fundamental challenge in deep learning: optimizing weight matrices through novel mathematical frameworks. The work extends existing optimization theory by moving beyond spectral norms, which have dominated recent approaches like the Muon update algorithm. By introducing the Fanion family grounded in Ky Fan norm duals, the researchers create a more flexible optimization toolkit that connects previously developed methods including ν-SAM and Dion under a unified theoretical umbrella.

The significance lies in how modern deep learning relies heavily on matrix optimization for training neural networks efficiently. The Muon algorithm gained attention for its simplicity and effectiveness, but this work demonstrates that alternative norm structures can achieve comparable or superior results. The creation of convex combination variants (F-Fanion and S-Fanion families) represents a methodological advance—allowing practitioners to blend different optimization strategies rather than committing to single approaches.

For AI practitioners and research communities, these findings suggest that spectral norms may be unnecessarily restrictive. F-Muon and S-Muon's consistent performance across diverse experimental settings indicates genuine algorithmic improvements rather than task-specific advantages. The synthetic convex problem outperformance hints at potential theoretical advantages in particular problem classes.

Looking forward, the practical impact depends on implementation availability and computational efficiency comparisons. If F-Muon and S-Muon prove computationally competitive with existing methods, they could see adoption in production deep learning frameworks. The theoretical connections drawn between previously disparate algorithms also open avenues for discovering new optimization approaches grounded in matrix norm theory.

• →Fanion family algorithms extend matrix optimization beyond spectral norms using Ky Fan norm duals
• →F-Muon and S-Muon variants match or exceed Muon performance across diverse benchmark tasks
• →The work unifies theoretical connections between Muon, ν-SAM, and Dion optimization methods
• →Convex combination approach enables flexible blending of different optimization strategies
• →Empirical results suggest potential advantages on specific problem classes like smooth convex problems