One Size Does Not Fit All: Curvature-Aware Adaptive Perturbation for SAM

Alessia Rossi; Felipe Campelo; Gabriel L. Oliveira

cs.LGOct 14, 2026

One Size Does Not Fit All: Curvature-Aware Adaptive Perturbation for SAM

Authors: Alessia Rossi, Felipe Campelo, Gabriel L. Oliveira

Organizations: School of Engineering Mathematics and Technology, University of Bristol, Bristol, Lovelace Building, Ada, Bristol, BS8 1TW, UK

Abstract

Poor generalisation in deep neural networks is frequently associated with convergence to sharp minima. Sharpness-Aware Minimisation (SAM) improves generalisation of deep neural networks by minimising a combination of the training loss and the local sharpness of the loss landscape, biasing convergence towards flat minima. However, SAM relies on a fixed perturbation radius $ρ$ that ignores local curvature, leading to inefficient behaviour: the perturbation may be too small to escape poor basins defined by sharp local minima, or too large to allow convergence to solutions with good loss-sharpness tradeoff. We reinterpret $ρ$ as the intensity of the perturbation signal and propose Curvature-Aware Adaptive Perturbation for SAM (CAP-SAM), a geometry-aware method that modulates the perturbation radius based on a sharpness ratio, enabling a more effective search for robust minima on the loss landscape. We prove that CAP-SAM preserves the same convergence rate as standard SAM in non-convex stochastic settings. Extensive experiments indicate that our approach consistently outperforms SAM and its most prominent variants across diverse network architectures on CIFAR-10, CIFAR-100, and ImageNet.

Figures & tables

Figure 1: Curvature-aware perturbation radius scheduling. The top panel illustrates how our method adjusts the perturbation radius based on the sharpness characteristics of the loss landscape. The bottom panel shows how

\rho_{t}

evolves: when in a sharp region,

\rho_{t}

expands by a factor of

\gamma

to look further ahead and escape the sharp basin; when in a flat region,

\rho_{t}

contracts by the same factor, so the optimizer stops perturbing aggressively and settles into the local minimum.

Panel of Figure 2 — Figure 2: Illustration of SAM perturbation directions in flat and sharp regions of the loss landscape.

Panel of Figure 3 — Figure 3: PyramidNet-110 on CIFAR-100. (a) Training loss $\mathcal{L}(\mathcal{D};\,w_{t})$ and (b) test accuracy (top-1) for CAP-SAM, SAM, and SGD.

	SAM		ASAM		Fisher SAM		VaSSO
Model	Const.	Dynam.	Const.	Dynam.	Const.	Dynam.	Const.	Dynam.
VGG-11-BN	$93.82_{\pm 0.05}$	$\mathbf{94.14_{\pm 0.09}}$	$93.47_{\pm 0.04}$	$\mathbf{94.13_{\pm 0.27}}$	$93.60_{\pm 0.09}$	$\mathbf{94.14_{\pm 0.15}}$	$93.93_{\pm 0.13}$	$\mathbf{94.17_{\pm 0.14}}$
ResNet-18	$96.58_{\pm 0.10}$	$\mathbf{96.66_{\pm 0.03}}$	$96.33_{\pm 0.09}$	$\mathbf{96.91_{\pm 0.08}}$	$96.62_{\pm 0.07}$	$\mathbf{96.79_{\pm 0.06}}$	$96.60_{\pm 0.08}$	$\mathbf{96.74_{\pm 0.23}}$
WRN-28-10	$97.32_{\pm 0.11}$	$\mathbf{97.47_{\pm 0.06}}$	$97.15_{\pm 0.05}$	$\mathbf{97.56_{\pm 0.07}}$	$97.46_{\pm 0.06}$	$\mathbf{97.48_{\pm 0.10}}$	$97.46_{\pm 0.09}$	$97.16_{\pm 0.08}$
PyramidNet-110	$97.58_{\pm 0.02}$	$\mathbf{97.70_{\pm 0.05}}$	$97.56_{\pm 0.11}$	$\mathbf{97.76_{\pm 0.03}}$	$97.59_{\pm 0.07}$	$\mathbf{97.77_{\pm 0.06}}$	$97.64_{\pm 0.09}$	$\mathbf{97.76_{\pm 0.05}}$

Table 1: Top-1 test accuracy (%) on CIFAR-10 with Cutout augmentation.

Panel of Figure 4 — Figure 4: Effect of hyperparameters with ResNet-18 on CIFAR-10.

	SAM		ASAM		Fisher SAM		VaSSO
Model	Const.	Dynam.	Const.	Dynam.	Const.	Dynam.	Const.	Dynam.
VGG-11-BN	$73.59_{\pm 0.10}$	$\mathbf{74.15_{\pm 0.14}}$	$68.53_{\pm 0.15}$	$\mathbf{73.46_{\pm 0.09}}$	$73.82_{\pm 0.31}$	$\mathbf{74.38_{\pm 0.13}}$	$74.02_{\pm 0.07}$	$\mathbf{74.23_{\pm 0.19}}$
ResNet-18	$80.52_{\pm 0.37}$	$\mathbf{81.10_{\pm 0.08}}$	$78.91_{\pm 0.27}$	$\mathbf{81.03_{\pm 0.12}}$	$80.82_{\pm 0.14}$	$\mathbf{81.12_{\pm 0.14}}$	$80.70_{\pm 0.21}$	$\mathbf{81.29_{\pm 0.24}}$
WRN-28-10	$84.64_{\pm 0.10}$	$\mathbf{85.10_{\pm 0.16}}$	$83.88_{\pm 0.27}$	$\mathbf{85.03_{\pm 0.21}}$	$84.47_{\pm 0.06}$	$\mathbf{84.94_{\pm 0.16}}$	$84.77_{\pm 0.07}$	$\mathbf{84.87_{\pm 0.28}}$
PyramidNet-110	$85.60_{\pm 0.11}$	$\mathbf{86.49_{\pm 0.11}}$	$83.72_{\pm 0.09}$	$\mathbf{86.90_{\pm 0.18}}$	$85.55_{\pm 0.14}$	$\mathbf{86.64_{\pm 0.23}}$	$85.85_{\pm 0.09}$	$\mathbf{86.89_{\pm 0.35}}$

Table 2: Top-1 test accuracy (%) on CIFAR-100 with Cutout augmentation.

Method		IMAGENET-
Method		1K	Real	V2	R	Sketch
SGD	—	$76.72_{\pm 0.09}$	$83.02_{\pm 0.02}$	$64.40_{\pm 0.07}$	$19.95_{\pm 0.12}$	$20.56_{\pm 0.20}$
SAM	Const.	$77.45_{\pm 0.088}$	$83.84_{\pm 0.08}$	$65.34_{\pm 0.16}$	$20.81_{\pm 0.34}$	$22.12_{\pm 0.17}$
SAM	Dynam.	$\mathbf{77.65_{\pm 0.11}}$	$\mathbf{84.03_{\pm 0.07}}$	$\mathbf{65.53_{\pm 0.18}}$	$\mathbf{20.95_{\pm 0.07}}$	$21.90_{\pm 0.18}$
GSAM	Const.	$77.37_{\pm 0.07}$	$83.83_{\pm 0.07}$	$65.42_{\pm 0.18}$	$20.93_{\pm 0.25}$	$21.26_{\pm 0.51}$
GSAM	Dynam.	$\mathbf{77.65_{\pm 0.15}}$	$\mathbf{83.98_{\pm 0.09}}$	$\mathbf{65.54_{\pm 0.22}}$	$\mathbf{21.14_{\pm 0.09}}$	$\mathbf{22.24_{\pm 0.16}}$

Table 3: Top-1 accuracy (%) on ImageNet and robustness datasets for ResNet-50.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 5 — Figure 5: Investigation of the SAM perturbation direction near convergence. (a) Away from convergence, $\nabla f(w_{t})$ dominates and the perturbation $\varepsilon_{\mathrm{SAM}}$ remains reasonably aligned with the true gradient direction. (b) Near a stationary point, $\nabla f(w_{t})\approx 0$ and $g_{t}(w_{t})\approx\xi$ , the perturbation direction becomes increasingly dominated by stochastic gradient noise making the perturbation uninformative about the local loss curvature Li et al. (2024) .

Factor	degrees-of-freedom	F-value	p-value	Mean acc. gain ( $\%$ )	$CI_{95}$
Factor	degrees-of-freedom	F-value	p-value	(dynam. - const.)	$CI_{95}$
Variant	$1$	$19.51$	$\mathbf{1.83\times 10^{-4}}$	0.23	$[0.12,0.33]$
Method	$3$	$0.451$	0.72	—	—
Model	$3$	$1107.6$	$\mathbf{7.42\times 10^{-26}}$	—	—
Residuals	$24$	—	—	—	—

Appendix

Table 4: ANOVA table and effect size estimation for the CIFAR-10 test. Statistically significant effects at the

95\%

confidence level are shown in boldface

Factor	degrees-of-freedom	F-value	p-value	Mean acc. gain ( $\%$ )	$CI_{95}$
Factor	degrees-of-freedom	F-value	p-value	(dynam. - const.)	$CI_{95}$
Variant	$1$	$13.32$	$\mathbf{1.27\times 10^{-3}}$	1.14	$[0.50,1.78]$
Method	$3$	$4.38$	$\mathbf{0.013}$	—	—
Model	$3$	$335.3$	$\mathbf{1.02\times 10^{-19}}$	—	—
Residuals	$24$	—	—	—	—

Appendix

Table 5: ANOVA table and effect size estimation for the CIFAR-100 test. Statistically significant effects at the

95\%

confidence level are shown in boldface

Factor	degrees-of-freedom	F-value	p-value	Mean acc. gain ( $\%$ )	$CI_{95}$
Factor	degrees-of-freedom	F-value	p-value	(dynam. - const.)	$CI_{95}$
Variant	$1$	$13.72$	$\mathbf{1.24\times 10^{-3}}$	0.31	$[0.14,0.48]$
Method	$2$	$1.41$	0.27	—	—
Version	$4$	$1.1\times 10^{5}$	$\mathbf{7.76\times 10^{-47}}$	—	—
Residuals	$24$	—	—	—	—

Appendix

Table 6: ANOVA table and effect size estimation for the first ImageNet test (Variant

\times

Method

\times

Version). Statistically significant effects at the

95\%

confidence level are shown in boldface

Figure 6: Mean accuracy gains of different Method-variant combinations against the SGD baseline, after removing the dataset Version effects. Using SAM-based approaches result in gains over the SGD baseline in all cases. Moreover, dynamic

\rho

results in higher gains when compared to constant

\rho

across all three methods tested.

Panel of Figure 7 — Figure 7: Effect of hyperparameters on top-1 accuracy of CAP-SAM with ResNet-18 on CIFAR-10.

Panel of Figure 8 — Figure 8: Training loss $\mathcal{L}(\mathcal{D};\,w_{t})$ of CAP-SAM as a function of the number of epochs, comparing CIFAR-10 and CIFAR-100 across four architectures: (a) ResNet-18,(b)WRN-28-10,(c) VGG-11-BN and (d) PyramidNet-110.

Panel of Figure 9 — Figure 9: Training loss $\mathcal{L}(\mathcal{D};\,w_{t})$ as a function of the number of epochs on CIFAR-100, comparing CAP-SAM, SAM, and SGD across four architectures: (a) ResNet-18, (b) WRN-28-10, (c) VGG-11-BN and (d) PyramidNet-110.

Panel of Figure 10 — Figure 10: Test top-1 accuracy as a function of the number of epochs on CIFAR-100, comparing CAP-SAM, SAM, and SGD across four architectures: (a) ResNet-18, (b) WRN-28-10, (c) VGG-11-BN and (d) PyramidNet-110.

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Sharpness-Aware Minimization (SAM) improves generalization by minimizing the worst-case loss within a fixed parameter-space radius neighborhood. SAM and its variants mainly rely on a first-order linearized surrogate, while flat minima are inherently a second-order (curvature) notion.We revisit this mismatch and propose Loss-Equated SAM (LE-SAM), which inverts the traditional SAM mechanism that fixed perturbation radius with a fixed loss-space budget,effectively removing gradient-norm-dominated learning signals and shifting optimization toward curvature-dominated terms. Extensive experiments across diverse benchmarks and tasks demonstrate the strong generalization ability of LESAM that consistently outperforms SAM and even its variants, achieving the state-of-the-art performance.

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