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

arXiv: https://arxiv.org/abs/2610.17285 · HTML: https://mlsanity.com/paper/2610.17285/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 40 references that are in the map.
- First posted on arXiv 12 Oct 2026.

## Builds on (most cited first, 30 of 30)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [SigLIP 2: Multilingual Vision-Language Encoders with Improved Semantic Understanding, Localization, and Dense Features](https://mlsanity.com/paper/2502.14786/) (2025)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [signSGD: Compressed Optimisation for Non-Convex Problems](https://mlsanity.com/paper/1802.04434/) (2018)
- [GLM-4.5: Agentic, Reasoning, and Coding (ARC) Foundation Models](https://mlsanity.com/paper/2508.06471/) (2025)
- [Muon is Scalable for LLM Training](https://mlsanity.com/paper/2502.16982/) (2025)
- [Shampoo: Preconditioned Stochastic Tensor Optimization](https://mlsanity.com/paper/1802.09568/) (2018)
- [SmolLM2: When Smol Goes Big -- Data-Centric Training of a Small Language Model](https://mlsanity.com/paper/2502.02737/) (2025)
- [SOAP: Improving and Stabilizing Shampoo using Adam](https://mlsanity.com/paper/2409.11321/) (2024)
- [Training Deep Learning Models with Norm-Constrained LMOs](https://mlsanity.com/paper/2502.07529/) (2025)
- [Primer: Searching for Efficient Transformers for Language Modeling](https://mlsanity.com/paper/2109.08668/) (2021)
- [Why Transformers Need Adam: A Hessian Perspective](https://mlsanity.com/paper/2402.16788/) (2024)
- [Fantastic Pretraining Optimizers and Where to Find Them](https://mlsanity.com/paper/2509.02046/) (2025)
- [Understanding Gradient Orthogonalization for Deep Learning via Non-Euclidean Trust-Region Optimization](https://mlsanity.com/paper/2503.12645/) (2025)
- [NorMuon: Making Muon more efficient and scalable](https://mlsanity.com/paper/2510.05491/) (2025)
- [Practical Efficiency of Muon for Pretraining](https://mlsanity.com/paper/2505.02222/) (2025)
- [Benchmarking Optimizers for Large Language Model Pretraining](https://mlsanity.com/paper/2509.01440/) (2025)
- [AdaMuon: Adaptive Muon Optimizer](https://mlsanity.com/paper/2507.11005/) (2025)
- [Muon Outperforms Adam in Tail-End Associative Memory Learning](https://mlsanity.com/paper/2509.26030/) (2025)
- [The Geometry of Sign Gradient Descent](https://mlsanity.com/paper/2002.08056/) (2020)
- [When do spectral gradient updates help in deep learning?](https://mlsanity.com/paper/2512.04299/) (2025)
- [The Sharpness Disparity Principle in Transformers for Accelerating Language Model Pre-Training](https://mlsanity.com/paper/2502.19002/) (2025)
- [Convergence Bound and Critical Batch Size of Muon Optimizer](https://mlsanity.com/paper/2507.01598/) (2025)
- [Adam Exploits ℓ_∞-geometry of Loss Landscape via Coordinate-wise Adaptivity](https://mlsanity.com/paper/2410.08198/) (2024)
- [Hyperparameter Transfer Enables Consistent Gains of Matrix-Preconditioned Optimizers Across Scales](https://mlsanity.com/paper/2512.05620/) (2025)
- [What Really Matters in Matrix-Whitening Optimizers?](https://mlsanity.com/paper/2510.25000/) (2025)
- [Preconditioned Norms: A Unified Framework for Steepest Descent, Quasi-Newton and Adaptive Methods](https://mlsanity.com/paper/2510.10777/) (2025)
- [Clarifying Shampoo: Adapting Spectral Descent to Stochasticity and the Parameter Trajectory](https://mlsanity.com/paper/2602.09314/) (2026)
- [Adaptive Batch Sizes Using Non-Euclidean Gradient Noise Scales for Stochastic Sign and Spectral Descent](https://mlsanity.com/paper/2602.03001/) (2026)

## Abstract (arXiv)

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.