Mousse: Rectifying the Geometry of Muon with Curvature-Aware Preconditioning

TL;DR

Mousse is a new optimizer that improves spectral optimization for deep neural networks by adapting to the curvature spectrum, outperforming Muon with fewer training steps.

Contribution

We introduce Mousse, a curvature-aware optimizer that combines spectral methods with second-order preconditioning using Shampoo's structural estimation.

Findings

Mousse reduces training steps by approximately 12% compared to Muon.

Mousse demonstrates improved stability and convergence on language models from 160M to 800M parameters.

Empirical results show negligible additional computational overhead.

Abstract

Recent advances in spectral optimization, notably Muon, have demonstrated that constraining update steps to the Stiefel manifold can significantly accelerate training and improve generalization. However, Muon implicitly assumes an isotropic optimization landscape, enforcing a uniform spectral update norm across all eigen-directions. We argue that this "egalitarian" constraint is suboptimal for Deep Neural Networks, where the curvature spectrum is known to be highly heavy-tailed and ill-conditioned. In such landscapes, Muon risks amplifying instabilities in high-curvature directions while limiting necessary progress in flat directions. In this work, we propose \textbf{Mousse} (\textbf{M}uon \textbf{O}ptimization \textbf{U}tilizing \textbf{S}hampoo's \textbf{S}tructural \textbf{E}stimation), a novel optimizer that reconciles the structural stability of spectral methods with the geometric adaptivity of second-order preconditioning. Instead of applying Newton-Schulz orthogonalization directly to the momentum matrix, Mousse operates in a whitened coordinate system induced by Kronecker-factored statistics (derived from Shampoo). Mathematically, we formulate Mousse as the solution to a spectral steepest descent problem constrained by an anisotropic trust region, where the optimal update is derived via the polar decomposition of the whitened gradient. Empirical results across language models ranging from 160M to 800M parameters demonstrate that Mousse consistently outperforms Muon, achieving around $\sim$12\% reduction in training steps with negligible computational overhead.