# Hide & Seek: Transformer Symmetries Obscure Sharpness & Riemannian Geometry Finds It

arXiv: https://arxiv.org/abs/2505.05409 · HTML: https://mlsanity.com/paper/2505.05409/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 36 references that are in the map.
- First posted on arXiv 8 May 2025.

## Builds on (most cited first, 22 of 22)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [A Broad-Coverage Challenge Corpus for Sentence Understanding through Inference](https://mlsanity.com/paper/1704.05426/) (2017)
- [Reliable evaluation of adversarial robustness with an ensemble of diverse parameter-free attacks](https://mlsanity.com/paper/2003.01690/) (2020)
- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [Model soups: averaging weights of multiple fine-tuned models improves accuracy without increasing inference time](https://mlsanity.com/paper/2203.05482/) (2022)
- [Weight Normalization: A Simple Reparameterization to Accelerate Training of Deep Neural Networks](https://mlsanity.com/paper/1602.07868/) (2016)
- [Sharp Minima Can Generalize For Deep Nets](https://mlsanity.com/paper/1703.04933/) (2017)
- [Kernel and Rich Regimes in Overparametrized Models](https://mlsanity.com/paper/2002.09277/) (2020)
- [Penalizing Gradient Norm for Efficiently Improving Generalization in Deep Learning](https://mlsanity.com/paper/2202.03599/) (2022)
- [Implicit Bias of SGD for Diagonal Linear Networks: a Provable Benefit of Stochasticity](https://mlsanity.com/paper/2106.09524/) (2021)
- [A Modern Look at the Relationship between Sharpness and Generalization](https://mlsanity.com/paper/2302.07011/) (2023)
- [BERTs of a feather do not generalize together: Large variability in generalization across models with similar test set performance](https://mlsanity.com/paper/1911.02969/) (2019)
- [Relative Flatness and Generalization](https://mlsanity.com/paper/2001.00939/) (2020)
- [In Search of Robust Measures of Generalization](https://mlsanity.com/paper/2010.11924/) (2020)
- [LoRA Done RITE: Robust Invariant Transformation Equilibration for LoRA Optimization](https://mlsanity.com/paper/2410.20625/) (2024)
- [The Geometry of Neural Nets' Parameter Spaces Under Reparametrization](https://mlsanity.com/paper/2302.07384/) (2023)
- [A Scale Invariant Flatness Measure for Deep Network Minima](https://mlsanity.com/paper/1902.02434/) (2019)
- [The Uncanny Valley: Exploring Adversarial Robustness from a Flatness Perspective](https://mlsanity.com/paper/2405.16918/) (2024)
- [On Geometric Connections of Embedded and Quotient Geometries in Riemannian Fixed-rank Matrix Optimization](https://mlsanity.com/paper/2110.12121/) (2021)

## Built on by (most cited first, 5 of 5)

- [Symmetry in Neural Network Parameter Spaces](https://mlsanity.com/paper/2506.13018/) (2025)
- [Generalizing the Geometry of Model Merging Through Frechet Averages](https://mlsanity.com/paper/2604.27155/) (2026)
- [Quantifying and Optimizing Simplicity via Polynomial Representations](https://mlsanity.com/paper/2605.29823/) (2026)
- [A Link between Shock-wave Theory and Symmetry-reduced Stochastic Gradient Descent for Artificial Neural Networks](https://mlsanity.com/paper/2606.18303/) (2026)
- [Effects of width-dependent model hyperparameters and ℓ2-regularization on the loss landscape of two-layer ReLU networks](https://mlsanity.com/paper/2607.16720/) (2026)

## Abstract (arXiv)

The concept of sharpness has been successfully applied to traditional architectures like MLPs and CNNs to predict their generalization. For transformers, however, recent work reported weak correlation between flatness and generalization. We argue that existing sharpness measures fail for transformers, because they have much richer symmetries in their attention mechanism that induce directions in parameter space along which the network or its loss remain identical. We posit that sharpness must account fully for these symmetries, and thus we redefine it on a quotient manifold that results from quotienting out the transformer symmetries, thereby removing their ambiguities. Leveraging tools from Riemannian geometry, we propose a fully general notion of sharpness, in terms of a geodesic ball on the symmetry-corrected quotient manifold. In practice, we need to resort to approximating the geodesics. Doing so up to first order yields existing adaptive sharpness measures, and we demonstrate that including higher-order terms is crucial to recover correlation with generalization. We present results on diagonal networks with synthetic data, and show that our geodesic sharpness reveals strong correlation for real-world transformers on both text and image classification tasks.