Orthogonal Witness Control for Muon Optimization via Sigmoid Spectral Reshaping

arXiv:2609.33194v1 Announce Type: new Abstract: Matrix-valued optimizers such as Muon exploit the spectral structure of neural network updates through Newton--Schulz orthogonalization, but their near-flattening of the singular spectrum discards relative magnitude information across gradient modes. We introduce Soren (extbf{S}pectral extbf{O}rthogonal extbf{Re}shapiextbf{n}g), a matrix-valued optimizer that preserves the singular subspaces of the gradient while applying a bounded, monotone sigmoid transformation to its singular values. This smoothly compresses dominant modes without fully flattening the spectrum. We interpret Soren as a positive-definite preconditioned gradient method and establish convergence guarantees under relative smoothness and metric Polyak--{L}ojasiewicz geometry. To avoid explicit singular value decomposition, we further develop a finite-depth Soft Newton--Schulz (SNS) polynomial realization of the sigmoid spectral map and characterize how its spectral approximation affects the induced convergence geometry. Experiments across LLM pre-training, supervised fine-tuning, and direct preference optimization demonstrate the effectiveness and robustness of Soren against established optimizers.

Source: arXiv cs.LG | 2026-09-29