# Navigating Potholes with Geometry-Aware Sharpness Minimization

arXiv: https://arxiv.org/abs/2605.16134 · HTML: https://mlsanity.com/paper/2605.16134/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 29 references that are in the map.
- First posted on arXiv 15 May 2026.

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

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Improved Regularization of Convolutional Neural Networks with Cutout](https://mlsanity.com/paper/1708.04552/) (2017)
- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima](https://mlsanity.com/paper/1609.04836/) (2016)
- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [Sharp Minima Can Generalize For Deep Nets](https://mlsanity.com/paper/1703.04933/) (2017)
- [Shampoo: Preconditioned Stochastic Tensor Optimization](https://mlsanity.com/paper/1802.09568/) (2018)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [ASAM: Adaptive Sharpness-Aware Minimization for Scale-Invariant Learning of Deep Neural Networks](https://mlsanity.com/paper/2102.11600/) (2021)
- [Scaling Neural Machine Translation](https://mlsanity.com/paper/1806.00187/) (2018)
- [Spectral Norm Regularization for Improving the Generalizability of Deep Learning](https://mlsanity.com/paper/1705.10941/) (2017)
- [Deep Pyramidal Residual Networks](https://mlsanity.com/paper/1610.02915/) (2016)
- [Penalizing Gradient Norm for Efficiently Improving Generalization in Deep Learning](https://mlsanity.com/paper/2202.03599/) (2022)
- [Path-SGD: Path-Normalized Optimization in Deep Neural Networks](https://mlsanity.com/paper/1506.02617/) (2015)
- [Make Sharpness-Aware Minimization Stronger: A Sparsified Perturbation Approach](https://mlsanity.com/paper/2210.05177/) (2022)
- [Friendly Sharpness-Aware Minimization](https://mlsanity.com/paper/2403.12350/) (2024)
- [Enhancing Sharpness-Aware Optimization Through Variance Suppression](https://mlsanity.com/paper/2309.15639/) (2023)
- [Layerwise LQR for Geometry-Aware Optimization of Deep Networks](https://mlsanity.com/paper/2605.04230/) (2026)

## Built on by (most cited first, 1 of 1)

- [K"ahler landscapes for complex neural network descents and guarantees including a search and destroy of the Calabi-Yau manifold](https://mlsanity.com/paper/2608.19584/) (2026)

## Abstract (arXiv)

Sharpness-aware minimization (SAM) encourages flat minima by perturbing parameters along directions of high loss curvature, but treats all parameter directions uniformly, ignoring the underlying loss geometry. We introduce LLQR+SAM, which combines SAM with a learned preconditioner obtained from the recently proposed LLQR framework, a second-order method that recasts steepest descent as a layerwise linear-quadratic regulator problem. The preconditioner is updated sparsely and maintained as a slow exponential moving average, so it captures a smoothed, low-resolution picture of the loss landscape geometry. The SAM perturbation then operates on top of this learned geometry, probing curvature at a faster timescale. We show that this two-timescale structure is not merely a computational convenience: theoretically, the preconditioner amplifies the SAM escape signal in directions that are flat under the average geometry but locally sharp (potholes). Wide, flat basins, by contrast, remain stable. Empirically, LLQR+SAM gives consistent gains over both SAM and LLQR alone across standard vision and sequence modeling benchmarks, supporting the view that slow learned geometry and fast sharpness correction are genuinely complementary.