# Flatness-aware Curriculum Learning via Adversarial Difficulty

arXiv: https://arxiv.org/abs/2508.18726 · HTML: https://mlsanity.com/paper/2508.18726/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 31 references that are in the map.
- First posted on arXiv 26 Aug 2025.

## Builds on (most cited first, 21 of 21)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Benchmarking Neural Network Robustness to Common Corruptions and Perturbations](https://mlsanity.com/paper/1903.12261/) (2019)
- [Improved Baselines with Momentum Contrastive Learning](https://mlsanity.com/paper/2003.04297/) (2020)
- [Fine-Grained Visual Classification of Aircraft](https://mlsanity.com/paper/1306.5151/) (2013)
- [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)
- [Deep Hashing Network for Unsupervised Domain Adaptation](https://mlsanity.com/paper/1706.07522/) (2017)
- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [In Search of Lost Domain Generalization](https://mlsanity.com/paper/2007.01434/) (2020)
- [Deeper, Broader and Artier Domain Generalization](https://mlsanity.com/paper/1710.03077/) (2017)
- [Recognition in Terra Incognita](https://mlsanity.com/paper/1807.04975/) (2018)
- [SWAD: Domain Generalization by Seeking Flat Minima](https://mlsanity.com/paper/2102.08604/) (2021)
- [ASAM: Adaptive Sharpness-Aware Minimization for Scale-Invariant Learning of Deep Neural Networks](https://mlsanity.com/paper/2102.11600/) (2021)
- [PyHessian: Neural Networks Through the Lens of the Hessian](https://mlsanity.com/paper/1912.07145/) (2019)
- [Penalizing Gradient Norm for Efficiently Improving Generalization in Deep Learning](https://mlsanity.com/paper/2202.03599/) (2022)
- [Sharpness-Aware Gradient Matching for Domain Generalization](https://mlsanity.com/paper/2303.10353/) (2023)
- [On the Duality Between Sharpness-Aware Minimization and Adversarial Training](https://mlsanity.com/paper/2402.15152/) (2024)
- [Why is SAM Robust to Label Noise?](https://mlsanity.com/paper/2405.03676/) (2024)
- [Rethinking the Evaluation Protocol of Domain Generalization](https://mlsanity.com/paper/2305.15253/) (2023)

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

- [Temperature-Adaptive Transformed Teacher Matching](https://mlsanity.com/paper/2608.29099/) (2026)

## Abstract (arXiv)

Neural networks trained by empirical risk minimization often suffer from overfitting, especially to specific samples or domains, which leads to poor generalization. Curriculum Learning (CL) addresses this issue by selecting training samples based on the difficulty. From the optimization perspective, methods such as Sharpness-Aware Minimization (SAM) improve robustness and generalization by seeking flat minima. However, combining CL with SAM is not straightforward. In flat regions, both the loss values and the gradient norms tend to become uniformly small, which makes it difficult to evaluate sample difficulty and design an effective curriculum. To overcome this problem, we propose the Adversarial Difficulty Measure (ADM), which quantifies adversarial vulnerability by leveraging the robustness properties of models trained toward flat minima. Unlike loss- or gradient-based measures, which become ineffective as training progresses into flatter regions, ADM remains informative by measuring the normalized loss gap between original and adversarial examples. We incorporate ADM into CL-based training with SAM to dynamically assess sample difficulty. We evaluated our approach on image classification tasks, fine-grained recognition, and domain generalization. The results demonstrate that our method preserves the strengths of both CL and SAM while outperforming existing curriculum-based and flatness-aware training strategies.