# Stability Analysis of Sharpness-Aware Minimization

arXiv: https://arxiv.org/abs/2301.06308 · HTML: https://mlsanity.com/paper/2301.06308/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 54 references that are in the map.
- First posted on arXiv 16 Jan 2023.

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

- [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)
- [Averaging Weights Leads to Wider Optima and Better Generalization](https://mlsanity.com/paper/1803.05407/) (2018)
- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data](https://mlsanity.com/paper/1703.11008/) (2017)
- [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)
- [Stochastic Gradient Descent as Approximate Bayesian Inference](https://mlsanity.com/paper/1704.04289/) (2017)
- [ASAM: Adaptive Sharpness-Aware Minimization for Scale-Invariant Learning of Deep Neural Networks](https://mlsanity.com/paper/2102.11600/) (2021)
- [Train longer, generalize better: closing the generalization gap in large batch training of neural networks](https://mlsanity.com/paper/1705.08741/) (2017)
- [Three Factors Influencing Minima in SGD](https://mlsanity.com/paper/1711.04623/) (2017)
- [When Vision Transformers Outperform ResNets without Pre-training or Strong Data Augmentations](https://mlsanity.com/paper/2106.01548/) (2021)
- [Surrogate Gap Minimization Improves Sharpness-Aware Training](https://mlsanity.com/paper/2203.08065/) (2022)
- [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)
- [An Alternative View: When Does SGD Escape Local Minima?](https://mlsanity.com/paper/1802.06175/) (2018)
- [When Do Flat Minima Optimizers Work?](https://mlsanity.com/paper/2202.00661/) (2022)
- [Gradient Descent Can Take Exponential Time to Escape Saddle Points](https://mlsanity.com/paper/1705.10412/) (2017)
- [Do We Need Zero Training Loss After Achieving Zero Training Error?](https://mlsanity.com/paper/2002.08709/) (2020)
- [The Dynamics of Sharpness-Aware Minimization: Bouncing Across Ravines and Drifting Towards Wide Minima](https://mlsanity.com/paper/2210.01513/) (2022)
- [Adaptive Inertia: Disentangling the Effects of Adaptive Learning Rate and Momentum](https://mlsanity.com/paper/2006.15815/) (2020)

## Built on by (most cited first, 14 of 14)

- [The Crucial Role of Normalization in Sharpness-Aware Minimization](https://mlsanity.com/paper/2305.15287/) (2023)
- [Sharpness-Aware Minimization Efficiently Selects Flatter Minima Late in Training](https://mlsanity.com/paper/2410.10373/) (2024)
- [An SDE for Modeling SAM: Theory and Insights](https://mlsanity.com/paper/2301.08203/) (2023)
- [Practical Sharpness-Aware Minimization Cannot Converge All the Way to Optima](https://mlsanity.com/paper/2306.09850/) (2023)
- [Decentralized SGD and Average-direction SAM are Asymptotically Equivalent](https://mlsanity.com/paper/2306.02913/) (2023)
- [Differentially Private Sharpness-Aware Training](https://mlsanity.com/paper/2306.05651/) (2023)
- [Exploring the Effect of Multi-step Ascent in Sharpness-Aware Minimization](https://mlsanity.com/paper/2302.10181/) (2023)
- [Stabilizing Sharpness-aware Minimization Through A Simple Renormalization Strategy](https://mlsanity.com/paper/2401.07250/) (2024)
- [Flat Minima and Generalization: Insights from Stochastic Convex Optimization](https://mlsanity.com/paper/2511.03548/) (2025)
- [Towards Understanding The Calibration Benefits of Sharpness-Aware Minimization](https://mlsanity.com/paper/2505.23866/) (2025)
- [Sharpness-Aware Minimization Can Hallucinate Minimizers](https://mlsanity.com/paper/2509.21818/) (2025)
- [Convergence of optimizers implies eigenvalues filtering at equilibrium](https://mlsanity.com/paper/2510.09034/) (2025)
- [Dispelling the Curse of Singularities in Neural Network Optimizations](https://mlsanity.com/paper/2602.01308/) (2026)
- [On the Implicit Flatness Bias of Sharpness-Aware Minimization: A Linear Stability Analysis with Quantitative Hyperparameter Bounds](https://mlsanity.com/paper/2608.03197/) (2026)

## Abstract (arXiv)

Sharpness-aware minimization (SAM) is a training method that seeks to find flat minima in deep learning, resulting in state-of-the-art performance across various domains. Instead of minimizing the loss of the current weights, SAM minimizes the worst-case loss in its neighborhood in the parameter space. In this paper, we investigate the convergence instability of SAM near a saddle point. Using the qualitative theory of dynamical systems, we explain how SAM becomes stuck in the saddle point and theoretically prove that the saddle point can become an attractor under SAM dynamics. Additionally, we show that this convergence instability can also occur in stochastic dynamical systems by establishing the diffusion of SAM. We prove that SAM diffusion is worse than that of vanilla gradient descent in terms of saddle point escape. Finally, we demonstrate that often overlooked training tricks, momentum and batch-size, might be important to mitigate the convergence instability and achieve high generalization performance. Our theoretical and empirical results are thoroughly verified through experiments on several well-known optimization problems and benchmark tasks.