# Sharpness-Aware Minimization Can Hallucinate Minimizers

arXiv: https://arxiv.org/abs/2509.21818 · HTML: https://mlsanity.com/paper/2509.21818/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 39 of its 45 references that are in the map.
- First posted on arXiv 26 Sep 2025.

## Builds on (most cited first, 30 of 39)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (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)
- [Exploring Generalization in Deep Learning](https://mlsanity.com/paper/1706.08947/) (2017)
- [Loss Surfaces, Mode Connectivity, and Fast Ensembling of DNNs](https://mlsanity.com/paper/1802.10026/) (2018)
- [Fantastic Generalization Measures and Where to Find Them](https://mlsanity.com/paper/1912.02178/) (2019)
- [Entropy-SGD: Biasing Gradient Descent Into Wide Valleys](https://mlsanity.com/paper/1611.01838/) (2016)
- [ASAM: Adaptive Sharpness-Aware Minimization for Scale-Invariant Learning of Deep Neural Networks](https://mlsanity.com/paper/2102.11600/) (2021)
- [Towards Understanding Sharpness-Aware Minimization](https://mlsanity.com/paper/2206.06232/) (2022)
- [Efficient Sharpness-aware Minimization for Improved Training of Neural Networks](https://mlsanity.com/paper/2110.03141/) (2021)
- [Towards Efficient and Scalable Sharpness-Aware Minimization](https://mlsanity.com/paper/2203.02714/) (2022)
- [Sharpness-Aware Training for Free](https://mlsanity.com/paper/2205.14083/) (2022)
- [Fisher SAM: Information Geometry and Sharpness Aware Minimisation](https://mlsanity.com/paper/2206.04920/) (2022)
- [Regularizing Neural Networks via Adversarial Model Perturbation](https://mlsanity.com/paper/2010.04925/) (2020)
- [When Do Flat Minima Optimizers Work?](https://mlsanity.com/paper/2202.00661/) (2022)
- [How Does Sharpness-Aware Minimization Minimize Sharpness?](https://mlsanity.com/paper/2211.05729/) (2022)
- [Friendly Sharpness-Aware Minimization](https://mlsanity.com/paper/2403.12350/) (2024)
- [The Dynamics of Sharpness-Aware Minimization: Bouncing Across Ravines and Drifting Towards Wide Minima](https://mlsanity.com/paper/2210.01513/) (2022)
- [An Adaptive Policy to Employ Sharpness-Aware Minimization](https://mlsanity.com/paper/2304.14647/) (2023)
- [Improving Sharpness-Aware Minimization with Fisher Mask for Better Generalization on Language Models](https://mlsanity.com/paper/2210.05497/) (2022)
- [PLASTIC: Improving Input and Label Plasticity for Sample Efficient Reinforcement Learning](https://mlsanity.com/paper/2306.10711/) (2023)
- [SAM as an Optimal Relaxation of Bayes](https://mlsanity.com/paper/2210.01620/) (2022)
- [Enhancing Sharpness-Aware Optimization Through Variance Suppression](https://mlsanity.com/paper/2309.15639/) (2023)
- [Why Does Sharpness-Aware Minimization Generalize Better Than SGD?](https://mlsanity.com/paper/2310.07269/) (2023)
- [Normalization Layers Are All That Sharpness-Aware Minimization Needs](https://mlsanity.com/paper/2306.04226/) (2023)
- [The Crucial Role of Normalization in Sharpness-Aware Minimization](https://mlsanity.com/paper/2305.15287/) (2023)

## Abstract (arXiv)

Sharpness-Aware Minimization (SAM) is widely used to seek flatter minima -- often linked to better generalization. In its standard implementation, SAM updates the current iterate using the loss gradient evaluated at a point perturbed by distance ρ along the normalized gradient direction. We show that, for some choices of ρ, SAM can stall at points where this shifted (perturbed-point) gradient vanishes despite a nonzero original gradient, and therefore, they are not stationary points of the original loss. We call these points hallucinated minimizers, prove their existence under simple nonconvex landscape conditions (e.g., the presence of a local minimizer and a local maximizer), and establish sufficient conditions for local convergence of the SAM iterates to them. We corroborate this failure mode in neural network training and observe that it aligns with SAM's performance degradation often seen at large ρ. Finally, as a practical safeguard, we find that a short initial SGD warm-start before enabling SAM mitigates this failure mode and reduces sensitivity to the choice of ρ.