aiqbench analysis
This study provides a rigorous linear stability analysis of Sharpness-Aware Minimization, deriving quantitative bounds that link hyperparameters like batch size and learning rate directly to the flatness of minima. By validating these theoretical predictions across 900 models on CIFAR-100, the authors demonstrate how adjusting the perturbation radius can effectively promote flatter solutions without compromising training stability. The work also introduces Taylor-Locality Controlled SAM as a practical variant that dynamically adapts this radius based on approximation error to further optimize generalization performance.
On the Implicit Flatness Bias of Sharpness-Aware Minimization: A Linear Stability Analysis with Quantitative Hyperparameter Bounds
arXiv:2608.03197v1 Announce Type: new Abstract: Sharpness-Aware Minimization (SAM) improves generalization by seeking parameters whose loss is robust to local adversarial perturbations, but the quantitative mechanism underlying its implicit bias toward flat minima remains unclear. In particular, the perturbation radius $\rho$ is typically treated as an isolated tuning parameter, despite defining the neighborhood in which SAM measures sharpness. We analyze mini-batch SAM near an interpolating minimum through linear stability. Under local linearization and gradient-noise alignment assumptions, we prove that every linearly stable minimum satisfies $\lambda_{\max}\leq\sqrt[3]{b\Gamma/(2\rho\eta^2)}$, where $\lambda_{\max}$ is the largest Hessian eigenvalue, $b$ is the batch size, $\eta$ is the learning rate, and $\Gamma$ bounds the gradient norm. The bound quantitatively characterizes SAM's implicit flatness bias: holding the other quantities fixed, a smaller batch size, a larger learning rate, or a larger radius restricts linearly stable SAM to flatter minima. It also exposes a necessary trade-off: $\rho$ should be large enough to promote flatness, yet remain local enough to preserve the approximation and stable training. We validate this prediction in a controlled study of 900 models on CIFAR-100 with ResNet-18 and VGG-19, where increasing $\rho$ is consistently associated with a smaller largest Hessian eigenvalue across batch-size and learning-rate settings. Finally, we instantiate the analysis in Taylor-Locality Controlled SAM (TLC-SAM), which adjusts $\rho$ using the observed Taylor-approximation error and further reduces the top Hessian eigenvalue relative to fixed-radius SAM. Our results provide quantitative hyperparameter bounds and a stability--locality perspective for analyzing and designing SAM variants.
Abstract:Sharpness-Aware Minimization (SAM) improves generalization by seeking parameters whose loss is robust to local adversarial perturbations, but the quantitative mechanism underlying its implicit bias toward flat minima remains unclear. In particular, the perturbation radius $\rho$ is typically treated as an isolated tuning parameter, despite defining the neighborhood in which SAM measures sharpness. We analyze mini-batch SAM near an interpolating minimum through linear stability. Under local linearization and gradient-noise alignment assumptions, we prove that every linearly stable minimum satisfies $\lambda_{\max}\leq\sqrt[3]{b\Gamma/(2\rho\eta^2)}$, where $\lambda_{\max}$ is the largest Hessian eigenvalue, $b$ is the batch size, $\eta$ is the learning rate, and $\Gamma$ bounds the gradient norm. The bound quantitatively characterizes SAM's implicit flatness bias: holding the other quantities fixed, a smaller batch size, a larger learning rate, or a larger radius restricts linearly stable SAM to flatter minima. It also exposes a necessary trade-off: $\rho$ should be large enough to promote flatness, yet remain local enough to preserve the approximation and stable training. We validate this prediction in a controlled study of 900 models on CIFAR-100 with ResNet-18 and VGG-19, where increasing $\rho$ is consistently associated with a smaller largest Hessian eigenvalue across batch-size and learning-rate settings. Finally, we instantiate the analysis in Taylor-Locality Controlled SAM (TLC-SAM), which adjusts $\rho$ using the observed Taylor-approximation error and further reduces the top Hessian eigenvalue relative to fixed-radius SAM. Our results provide quantitative hyperparameter bounds and a stability--locality perspective for analyzing and designing SAM variants.
Machine Learning (cs.LG)
arXiv:2608.03197 [cs.LG]
(or arXiv:2608.03197v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2608.03197 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.
Full text under the publisher's open licence · 332 words
Related coverage
Conflicting Biases at the Edge of Stability: Norm versus Sharpness Regularization
arXiv · Jun 8, 2026
Symmetry-quotient Flatness and Generalization
arXiv · Sep 30, 2026
GEM Training: How Meta Doubled the Efficiency of Its LLM-Scale Ads Foundation Model
Meta Engineering · Aug 3, 2026
Do Flatter Minima Drive Better Generalization? An Algorithmic Separation in Grokking
arXiv · Oct 9, 2026