Title: On the Convergence of Two-Layer Kolmogorov-Arnold Networks with First-Layer Training
Primary area: learning theory
Keywords: Kolmogorov-Arnold Networks (KANs), Overparameterization, Neural Tangent Kernel (NTK), Gradient Descent

Abstract: Kolmogorov-Arnold Networks (KANs) have emerged as a promising alternative to traditional neural networks, offering enhanced interpretability based on the Kolmogorov-Arnold representation theorem. While their empirical success is growing, a theoretical understanding of their training dynamics remains nascent. This paper investigates the optimization of a two-layer KAN in the overparameterized regime, focusing on a simplified yet insightful setting where only the first-layer coefficients are trained via gradient descent.

Our main result establishes that, provided the network is sufficiently wide, this training method is guaranteed to converge to a global minimum and achieve zero training error. Furthermore, we derive a novel, fine-grained convergence rate that explicitly connects the optimization speed to the structure of the data labels through the eigenspectrum of the KAN Tangent Kernel (KAN-TK). Our analysis reveals a key advantage of this architecture: guaranteed convergence is achieved with a hidden layer width of $m=\mathcal{O}(n^2)$, a significant polynomial improvement over the $m=\mathcal{O}(n^6)$ requirement for classic two-layer neural networks. We validate our theoretical findings with numerical experiments that corroborate our predictions on convergence speed and the impact of label structure.