Which Geometry on Which Layer? Measured Sign-Spectral Assignment in Mixed-Optimizer Training

Abstract

Modern neural-network training uses mixed recipes, assigning AdamW-like or sign-based updates to embedding and output layers and spectral updates to hidden matrices. These assignments are usually fixed by layer type, without a clear criterion for selecting the appropriate geometry. We turn sign-versus-spectral assignment into a measurable, layerwise decision problem. For ideal memoryless steps, we derive a finite-batch ratio of conservative descent-lemma lower bounds for spectral and sign updates. Its closed form separates signal efficiency from a geometry-dependent global curvature factor. In the positive-signal regime, the same smoothness assumption gives two-sided bounds that determine which step has the larger conditional expected decrease when the ratio crosses explicit noise-dependent thresholds. We then derive a local directional quadratic score from gradient alignments and Hessian-vector products and use it in Generalized-Scion (G-Scion) to control layerwise routing. In a controlled input-rank experiment, including directional curvature matches the measured same-iterate ordering in all seven aggregate settings, whereas the curvature-free proxy selects the opposite geometry. On 31M GPT-2, the output score routes the output layer from spectral warmup to sign, and G-Scion improves validation loss over fixed Scion in memoryless and matched-momentum comparisons. ViT-B/16 results further show that a spectral edge assignment can outperform fixed Scion outside language modelling. These results suggest that optimizer geometry need not be hard-coded by layer type and are a first step toward understanding how it should be assigned.