Hybrid optimization framework for improving Kolmogorov-Arnold network in federated learning

doi:10.11772/j.issn.1001-9081.2025050536

Journal of Computer Applications ›› 2026, Vol. 46 ›› Issue (4): 1023-1033.DOI: 10.11772/j.issn.1001-9081.2025050536

• Artificial intelligence • Next Articles

Hybrid optimization framework for improving Kolmogorov-Arnold network in federated learning

Zhi JIANG¹, Xuebin CHEN¹(), Changyin LUO¹^,², Ziye ZHEN¹

^1.College of Sciences，North China University of Science and Technology，Tangshan Hebei 063210，China
^2.School of Mathematics and Statistics，Ningxia University，Yinchuan Ningxia 750021，China

Received:2025-05-16 Revised:2025-06-27 Accepted:2025-07-15 Online:2025-08-01 Published:2026-04-10
Contact: Xuebin CHEN
About author:JIANG Zhi， born in 2000， M. S. candidate. His research interests include federated learning， machine learning.
LUO Changyin， born in 1994， Ph. D. candidate. His research interests include federated learning， machine learning.
ZHEN Ziye， born in 2001， M. S. candidate. His research interests include radial basis function interpolation， multivariate spline， deep learning.
Supported by:
National Natural Science Foundation of China(U20A20179)

联邦学习中改进Kolmogorov-Arnold网络的混合优化框架

姜志¹, 陈学斌¹(), 罗长银¹^,², 甄子业¹

^1.华北理工大学理学院，河北唐山 063210
^2.宁夏大学数学与统计学院，银川 750021

通讯作者: 陈学斌
作者简介:姜志（2000—），男，山东青岛人，硕士研究生，CCF会员，主要研究方向：联邦学习、机器学习
罗长银（1994—），男，陕西安康人，博士研究生，CCF会员，主要研究方向：联邦学习、机器学习
甄子业（2001—），男，河北定州人，硕士研究生，主要研究方向：径向基函数插值、多元样条、深度学习。
基金资助:
国家自然科学基金资助项目(U20A20179)

Abstract

Abstract:

For addressing issues such as data heterogeneity， tendency of gradients to converge to local optimum， and high computational and communication overhead in federated learning， a hybrid training framework of “key edge screening-early-stopping genetic evolution-local fine-tuning” was developed for Kolmogorov-Arnold Network （KAN）， called KB-GA-KAN. First， key edges on each client were selected dynamically according to kernel function amplitude and activation sensitivity， and only the kernel coefficients of these edges were evolved genetically， enabling a global search for good initial solutions. Then， an early-stopping criterion was introduced， and collaborative optimization was achieved by combining the evolution with local Stochastic Gradient Descent （SGD）. Experimental results on five Non-Independent and Identically Distributed （Non-IID） datasets demonstrate that compared to KAN framework with pure gradient training， KB-GA-KAN has test accuracy raised by an average of 1.34%， and the number of convergence rounds lowered by 42%， and it improves the robustness of heterogeneous scenarios with a slight additional computational cost. Visual results of the kernel functions further confirm that KB-GA-KAN enhances model interpretability. It can be seen that KB-GA-KAN offers a new route to balance accuracy， convergence speed， and computational cost of efficient SGD KAN under privacy-restricted conditions.

Key words: federated learning, Kolmogorov-Arnold Network (KAN), Genetic Algorithm (GA), early stopping mechanism

摘要：

针对联邦学习中数据异质、梯度易陷入局部最优及计算?通信开销偏高等问题，面向Kolmogorov-Arnold网络（KAN）提出一种“关键边筛选?早停遗传进化?局部微调”的混合训练框架——KB-GA-KAN。首先依据核函数幅度和激活敏感度在客户端动态选取关键边，并仅对这些边的核系数进行遗传进化，全局搜索优良初始解；然后引入早停判据，结合进化与本地随机梯度下降（SGD）实现协同优化。在5个非独立同分布（Non-IID）数据集上的实验结果表明：相较于纯梯度训练的KAN，KB-GA-KAN在相同通信轮数下的测试准确率平均提高了1.34%，收敛轮数减少了42%，并以轻微的额外计算代价提升了异构场景的鲁棒性。核函数的可视化结果进一步验证了KB-GA-KAN对模型可解释性的促进作用。可见，KB-GA-KAN能为隐私受限条件下的高效随机梯度下降KAN训练提供兼顾准确率、收敛速度与计算成本的新途径。

关键词: 联邦学习, Kolmogorov-Arnold网络, 遗传算法, 早停机制

CLC Number:

TP309.2

Zhi JIANG, Xuebin CHEN, Changyin LUO, Ziye ZHEN. Hybrid optimization framework for improving Kolmogorov-Arnold network in federated learning[J]. Journal of Computer Applications, 2026, 46(4): 1023-1033.

姜志, 陈学斌, 罗长银, 甄子业. 联邦学习中改进Kolmogorov-Arnold网络的混合优化框架[J]. 《计算机应用》唯一官方网站, 2026, 46(4): 1023-1033.

Figures/Tables 11

Fig. 1 Problems of traditional RBF-KAN on two datasets

Fig. 2 Kolmogorov-Arnold network

Fig. 3 Overall framework of KB-GA-KAN

Tab. 1 Accuracies of different methods on different datasets

方法	Fashion-MNIST	SVHN	CIFAR-10
FedAvg	87.84±0.46	86.21±0.68	66.93±0.21
FedProx	87.97±0.28	86.35±0.43	67.01±0.03
SCAFFOLD	88.06±0.06	86.42±0.59	67.53±0.70
Fedl₁	88.31±0.33	86.43±0.61	67.20±0.73
FedDANE	72.81±1.26	67.69±2.31	62.27±3.23
MOON	87.84±0.46	84.03±0.72	69.35±0.11
KB-GA-KAN	88.34±0.13	87.02±0.54	69.71±0.15

Tab. 2 Accuracies of different methods with different β on two datasets

方法	MNIST				CIFAR-100
方法	$β = 0.05$	$β = 0.10$	$β = 0.50$	$β = 5.00$	$β = 0.05$	$β = 0.10$	$β = 0.50$	$β = 5.00$
FedAvg	97.67±0.23	97.80±0.08	98.45±0.06	98.58±0.10	25.17±0.49	25.82±0.61	25.97±0.66	26.07±0.58
FedProx	97.77±0.22	98.04±0.22	98.62±0.11	98.70±0.09	25.20±0.48	25.78±0.60	26.06±0.59	26.12±0.76
SCAFFOLD	97.53±0.67	97.69±0.60	98.56±0.13	98.70±0.09	24.68±0.25	24.98±0.52	25.67±0.84	26.12±0.76
Fedl₁	97.73±0.10	97.86±0.01	98.52±0.11	98.64±0.08	26.06±0.11	26.31±0.55	26.40±0.14	26.63±0.16
FedDANE	96.49±1.96	96.92±0.83	97.13±0.54	97.91±0.13	22.32±0.69	22.43±0.25	22.79±0.38	22.93±0.23
MOON	97.41±0.23	97.75±0.46	98.58±0.09	98.69±0.05	24.65±0.53	25.72±0.67	25.81±0.81	26.24±0.64
KB-GA-KAN	97.79±0.18	97.91±0.39	98.79±0.08	98.85±0.07	26.57±0.52	26.83±0.25	26.09±0.14	27.02±0.09

Tab. 2 Accuracies of different methods with different β on two datasets

方法	MNIST				CIFAR-100
方法	$β = 0.05$	$β = 0.10$	$β = 0.50$	$β = 5.00$	$β = 0.05$	$β = 0.10$	$β = 0.50$	$β = 5.00$
FedAvg	97.67±0.23	97.80±0.08	98.45±0.06	98.58±0.10	25.17±0.49	25.82±0.61	25.97±0.66	26.07±0.58
FedProx	97.77±0.22	98.04±0.22	98.62±0.11	98.70±0.09	25.20±0.48	25.78±0.60	26.06±0.59	26.12±0.76
SCAFFOLD	97.53±0.67	97.69±0.60	98.56±0.13	98.70±0.09	24.68±0.25	24.98±0.52	25.67±0.84	26.12±0.76
Fedl₁	97.73±0.10	97.86±0.01	98.52±0.11	98.64±0.08	26.06±0.11	26.31±0.55	26.40±0.14	26.63±0.16
FedDANE	96.49±1.96	96.92±0.83	97.13±0.54	97.91±0.13	22.32±0.69	22.43±0.25	22.79±0.38	22.93±0.23
MOON	97.41±0.23	97.75±0.46	98.58±0.09	98.69±0.05	24.65±0.53	25.72±0.67	25.81±0.81	26.24±0.64
KB-GA-KAN	97.79±0.18	97.91±0.39	98.79±0.08	98.85±0.07	26.57±0.52	26.83±0.25	26.09±0.14	27.02±0.09

Tab. 3 Communication efficiencies of different methods

方法	Fashion-MNIST		SVHN		CIFAR-10
方法	轮数	速度	轮数	速度	轮数	速度
FedAvg	50	1.00×	50	1.00×	50	1.00×
FedProx	49	1.02×	48	1.04×	49	1.02×
SCAFFOLD	47	1.06×	47	1.06×	45	1.11×
Fedl₁	48	1.04×	44	1.13×	47	1.06×
FedDANE	52	0.96×	64	0.78×	55	0.91×
MOON	36	1.38×	38	1.31×	16	3.13×
KB-GA-KAN	17	2.94×	21	2.38×	13	3.84×

Tab. 4 Number of communication rounds required to achieve target accuracy on two datasets

数据集	方法	E=10， B=32		E=10， B=128		E=5， B=32
数据集	方法	20%客户端	50%客户端	20%客户端	50%客户端	20%客户端	50%客户端
MNIST	FedAvg	50（1×）	50（1×）	50（1×）	50（1×）	50（1×）	50（1×）
	FedProx	49	37	48	39	55	46
	SCAFFOLD	47	29	47	30	49	34
	Fedl₁	43	27	45	31	47	33
	FedDANE	41	34	45	36	45	36
	MOON	36	29	40	30	41	37
	KB-GA-KAN	20	13	25	17	22	15
CIFAR-100	FedAvg	50（1×）	50（1×）	50（1×）	50（1×）	50（1×）	50（1×）
	FedProx	49	39	49	44	49	48
	SCAFFOLD	46	40	45	33	44	29
	Fedl₁	48	30	44	31	42	39
	FedDANE	42	36	47	40	39	35
	MOON	39	22	43	39	40	27
	KB-GA-KAN	23	14	28	24	25	16

Fig. 4 Accuracy convergence curves on different datasets

Tab. 5 Model accuracies and time costs under different early stopping thresholds

早停阈值	准确率/%	时间成本/s
10^-2	86.1	12.1
10^-3	86.7	13.6
10^-4	87.2	15.3
10^-5	87.7	17.1
10^-6	88.1	19.7
10^-7	88.3	23.4
10^-8	88.4	29.6

Tab. 6 Communication data sizes of different methods

方法	轻量KAN-Lite	中型KAN-Base	重型KAN-Deep
FedAvg/Prox/MOON	51.2	78.4	102.4
Fedl₁	50.1	71.6	91.5
SCAFFOLD	76.8	96.4	142.0
FedDANE	153.6	233.6	293.8
KB-GA-KAN	29.6	45.1	60.2

Tab. 7 Ablation experimental results of KB-GA-KAN core modules

方法	是否筛选关键边	GA优化范围	是否有早停机制	收敛轮数	准确率/%	时间开销/s
BP-KAN	否	GA不参与	否	48	86.42±0.29	13.70
全参数GA-KAN	否	全参数	否	56	87.71±0.08	35.90
无早停KB-GA-KAN	是	Key Edge	否	23	87.78±0.27	22.95
KB-GA-KAN	是	Key Edge	是	23	87.75±0.13	15.30

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Hybrid optimization framework for improving Kolmogorov-Arnold network in federated learning

联邦学习中改进Kolmogorov-Arnold网络的混合优化框架

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