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.