Taming Softmax at the Limits of Muon: A Unified Approach through Hilbert Geometry
Muon controls weight updates through the RMS movement of a linear layer's output. This implicitly treats the linear output as a proxy for the full module, including the nonlinearity that follows. This proxy can be misleading for softmax, where the RMS shift in activations does not faithfully measure the change in the output distribution. We propose Hilbert distance as a natural metric for bounding changes in these output distributions. Hilbert distance bounds changes in cross-entropy loss and limits probability redistribution. For softmax, it takes the form of the variation seminorm of the preactivation perturbation, allowing probability constraints to be expressed as parameter constraint...
| Authors | Aditya Somasundaram, Charles Guille-Escuret, Alexander Moreno et al. |
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| Primary category | cs.LG |
| Published | 13 October 2026 |
| Last updated at source | 13 October 2026 |
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