Which Geometry on Which Layer? Measured Sign–Spectral Assignmentin Mixed-Optimizer Training

Modern training of large neural networks now commonly uses mixed recipes, assigning AdamW-like or sign-based updates to embedding and output layers and spectral updates to hidden matrices. These assignments are fixed by layer type for the entire run and are largely inherited heuristics that must be re-tuned across architectures and training regimes, without a clear criterion for selecting the appropriate geometry, that is, the norm under which the steepest descent step is taken. We turn the sign-versus-spectral assignment into a measurable, layerwise decision problem. For ideal memoryless steps, we derive the finite-batch geometry ratio $R\_\mathrm{LB}(W;B):=\mathcal{B}\_\mathrm{spec}(W)/\mathcal{B}\_\mathrm{sign}(W)$, where $\mathcal{B}\_\mathrm{spec}$ and $\mathcal{B}\_\mathrm{sign}$ are conservative descent-lemma lower bounds for the specified spectral and sign steps. 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 experimental setup, including directional curvature matches the measured same-iterate ordering in all seven aggregate settings, whereas the curvature-free proxy selects the opposite geometry. On $31$M 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 they are a first step toward understanding how it should be assigned.