FedNSAM:Consistency of Local and Global Flatness for Federated Learning

Junkang Liu; Fanhua Shang; Yuxuan Tian; Hongying Liu; Yuanyuan Liu

cs.LGFeb 27, 2026

FedNSAM:Consistency of Local and Global Flatness for Federated Learning

Authors: Junkang Liu, Fanhua Shang, Yuxuan Tian, Hongying Liu, Yuanyuan Liu

Organizations: College of Intelligence and Computing, Tianjin University, Tianjin, China · College of Management and Economics, Tianjin University, Tianjin, China · Medical College, Tianjin University; Peng Cheng Laboratory, Tianjin, China · School of Artificial Intelligence, Xidian University, Xian, China

Abstract

In federated learning (FL), multi-step local updates and data heterogeneity usually lead to sharper global minima, which degrades the performance of the global model. Popular FL algorithms integrate sharpness-aware minimization (SAM) into local training to address this issue. However, in the high data heterogeneity setting, the flatness in local training does not imply the flatness of the global model. Therefore, minimizing the sharpness of the local loss surfaces on the client data does not enable the effectiveness of SAM in FL to improve the generalization ability of the global model. We define the flatness distance to explain this phenomenon. By rethinking the SAM in FL and theoretically analyzing the flatness distance, we propose a novel FedNSAM algorithm that accelerates the SAM algorithm by introducing global Nesterov momentum into the local update to harmonize the consistency of global and local flatness. FedNSAM uses the global Nesterov momentum as the direction of local estimation of client global perturbations and extrapolation. Theoretically, we prove a tighter convergence bound than FedSAM by Nesterov extrapolation. Empirically, we conduct comprehensive experiments on CNN and Transformer models to verify the superior performance and efficiency of FedNSAM. The code is available at https://github.com/junkangLiu0/FedNSAM.

Figures & tables

Panel of Figure 1 — Figure 1. (a) and (b) show the global training loss surface of FedSAM ( Qu et al., 2022 ) under Dirichlet distributions with coefficients of 0.6 (low data heterogeneity) and 0.1 (high data heterogeneity) on CIFAR-100 with ResNet-18. (c) shows the training loss surface of our FedNSAM with Dirichlet-0.1. FedSAM can search global flat minima in low data heterogeneity but fails in high data heterogeneity. (d) suggests that the flat region of the client is closer in low data heterogeneity setting. The global model falls within the flat region. (e) suggests that the flat region of the client is far in high data heterogeneity setting. The global model cannot fall within the flat region. (f) suggests that our FedNSAM draws the flat region between each client closer by alignment correction so that the global model falls within the flat region of each client. FedNSAM ’s global model finds flat minima in (c).

Panel of Figure 2 — Figure 2. Illustration of flatness distance (left) and global sharpness (right) during federated training on CIFAR100, using Dirichlet distributions with coefficients 0.1, 0.3, and 0.6 across 100 clients and a 10% participation rate. Test accuracies are 40.18%, 46.02%, and 47.83% for FedSAM with Dirichlet-0.1, 0.3, and 0.6 respectively, and 58.53% for FedNSAM with Dirichlet-0.1.

Research work	Perturbation Correction	Updating correction	Acceleration	Less Computation	Com Cost
FedSAM ( Qu et al., 2022 )	$\times$	$\times$	$\times$	$\times$	1 $\times$
MoFedSAM ( Qu et al., 2022 )	$\times$	✓	$\times$	$\times$	1 $\times$
FedGAMMA ( Dai et al., 2023 )	$\times$	✓	$\times$	$\times$	2 $\times$
FedSMOO ( Sun et al., 2023 )	✓	$\times$	$\times$	$\times$	2 $\times$
FedLESAM ( Fan et al., 2024 )	✓	$\times$	$\times$	✓	1 $\times$
FedNSAM (ours)	✓	✓	✓	✓	1 $\times$

Table 1. Summary of federated SAM-based algorithms for addressing data heterogeneity, focusing on perturbation correction, updating correction, acceleration, and extra computation introduced by SAM. Here, perturbation correction is a correction for

\boldsymbol{\delta}_{i,k}^{t}

, updating correction is a correction for

\nabla F_{i}(\boldsymbol{\theta}_{k+1/2}^{t})

, and Com Cost denotes the communication cost.

Panel of Figure 3 — Figure 3. (a) and (b) depict the local update procedures of FedSAM and FedNSAM , respectively. FedNSAM incorporates global Nesterov momentum into local sharpness-aware updates to improve alignment between local and global flatness.

CIFAR10

CIFAR100

Method

LeNet-5

VGG-11

ResNet-18

LeNet-5

VGG-11

ResNet-18

Acc.(%)

Rounds

Acc.(%)

Rounds

Acc.(%)

Rounds

Acc.(%)

Rounds

Acc.(%)

Rounds

Acc.(%)

Rounds

1000R

78%

1000R

80%

1000R

85%

1000R

52%

1000R

53%

1000R

55%

FedAvg

79.63\pm 0.27

574

84.14\pm 0.32

313

88.92\pm 0.24

542

41.15\pm 0.18

1000+

48.94\pm 0.39

1000+

54.25\pm 0.39

1000+

FedAvgM

81.15\pm 0.23

659

83.74\pm 0.11

794

89.05\pm 0.36

656

48.35\pm 0.29

1000+

51.94\pm 0.17

1000+

60.91\pm 0.37

516

SCAFFOLD

79.43\pm 0.33

799

86.94\pm 0.09

272

88.56\pm 0.21

543

50.21\pm 0.35

1000+

50.81\pm 0.28

1000+

54.12\pm 0.15

1000+

Table 2. Comparison of testing accuracy (%) on CIFAR10 and CIFAR100,

E=5

, Dirichlet-0.6 respectively.

Panel of Figure 4 — Figure 4. Convergence plots for FedNSAM and other baselines on Dirichlet-0.6 of CIFAR10 and CIFAR100 with ResNet-18.

Panel of Figure 5 — Figure 5. Convergence plots for FedNSAM with different $\lambda$ and $\rho$ , CIFAR100 datasets with ResNet-18.

Data Set	Tiny ImageNet
Model	Swin-Small		Swin-Base		ViT-Base
Param	50M		80M		88M
FLOPs	8.7G		15.4G		55.4G
	Acc	Rounds	Acc	Rounds	Acc	Rounds
Method	100R	68%	100R	68%	100R	68%
FedAvg	69.80 $\pm 0.21$	100+	67.41	100+	64.64	100+

Table 3. Comparision the accuracy of different Transformer models with Dirichlet-0.1, 5% participation.

	Participation Rate						Data Heterogeneity (Dirichlet- $\alpha$ )
Method	$p=2\%$		$p=5\%$		$p=10\%$		$\alpha=0.1$		$\alpha=0.3$		$\alpha=0.6$
	Acc	R	Acc	R	Acc	R	Acc	R	Acc	R	Acc	R
FedAvg	51.61 $\pm$ 0.35	1000+	51.53 $\pm$ 0.12	1000+	54.25 $\pm$ 0.39	1000+	45.81 $\pm$ 0.19	1000+	52.53 $\pm$ 0.15	1000+	54.25 $\pm$ 0.39	1000+
FedAvgM	15.72 $\pm$ 0.23	1000+	46.63 $\pm$ 0.33	1000+	60.91 $\pm$ 0.37	516	48.63 $\pm$ 0.40	1000+	58.84 $\pm$ 0.08	808	60.91 $\pm$ 0.37	516
SCAFFOLD	51.93 $\pm$ 0.29	1000+	53.64 $\pm$ 0.37	1000+	54.12 $\pm$ 0.28	1000+	49.02 $\pm$ 0.33	1000+	51.92 $\pm$ 0.25	1000+	54.12 $\pm$ 0.28	1000+
FedACG	54.01 $\pm$ 0.16	1000+	61.94 $\pm$ 0.26	506	61.92 $\pm$ 0.31	521	53.56 $\pm$ 0.34	1000+	60.87 $\pm$ 0.12	425	61.92 $\pm$ 0.31	521

Table 4. Comparison of testing accuracy (%) and convergence rounds on CIFAR100 (

E=5

) for different participation rates (left) and different data heterogeneity levels (right).

Panel of Figure 6 — Figure 6. Convergence plots of FedNSAM and baselines on CIFAR100 with ResNet-18 under different heterogeneity (a,b) and participation rates (c,d).

Panel of Figure 7 — Figure 7. Global training loss surfaces plots (a,b,c,d,e,f) and global testing loss surfaces plots (g,h,i,j,k,l) for FedNSAM and other baselines in different settings on Dirichlet-0.1 of CIFAR100 datasets with ResNet-18. FedNSAM could approach a more general and flat loss landscape which efficiently improves the generalization performance in FL.

	CIFAR100
Method	Dirichlet-0.1		Dirichlet-0.6
	Acc	Rounds	Acc	Rounds
	1000R	55%	1000R	55%
FedAvg	45.81 $\pm 0.35$	1000+	54.25 $\pm 0.19$	1000+
FedNSAM	58.53 $\pm 0.27$	695	66.04 $\pm 0.22$	316
SCAFFOLD	49.02 $\pm 0.29$	1000+	54.12 $\pm 0.40$	1000+

Table 5. Comparison different data heterogeneity of testing accuracy (%) of ablation on variants on CIFAR100.

	CIFAR100
Method	Dirichlet-0.1		Dirichlet-0.6
	Acc	Rounds	Acc	Rounds
	1000R	55%	1000R	55%
FedAvg	45.81 $\pm 0.32$	1000+	54.25 $\pm 0.15$	1000+
FedAvg+Nesterov	54.23 $\pm 0.28$	1000+	60.26 $\pm 0.39$	512
FedSAM	40.18 $\pm 0.27$	1000+	47.83 $\pm 0.09$	1000+

Table 6. Comparison different data heterogeneity of testing accuracy (%) of each algorithm with Nesterov on CIFAR100.

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Sharpness-aware minimization (SAM) is an effective method for improving the generalization of federated learning (FL) by steering local training toward flat minima. Under data heterogeneity, however, device-side SAM searches for locally flat basins that are incompatible with the flat region preferred by the global objective. We identify this structural failure mode as flatness incompatibility, which explains why improving local flatness alone may provide limited training and generalization improvement for the global model. We reveal that flatness incompatibility arises from data heterogeneity and the friendly adversary phenomenon, and is further amplified by local updates and partial device participation. To mitigate this issue, we propose Federated Learning with variance-suppressed sharpness-aware minimization (FedVSSAM), which constructs a variance-suppressed adjusted direction and uses it consistently in local flatness search, local descent, and global update. FedVSSAM anchors both perturbation and update directions to a more stable global direction, instead of correcting only an isolated local perturbation. We establish non-convex convergence guarantees of FedVSSAM and prove that the mean-square deviation between the adjusted direction and the global gradient is effectively controlled. Experiments demonstrate that FedVSSAM mitigates flatness incompatibility and outperforms the baselines across diverse FL settings.

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