Title: On the Rate of Convergence of Kolmogorov-Arnold Network Regression Estimators
Primary area: learning theory
Keywords: Kolmogorov-Arnold Network, convergence rate, minimax optimality, nonparametric regression, B-splines, sieve estimators

Abstract: Kolmogorov-Arnold Networks (KANs) offer a structured and interpretable framework for multivariate function approximation by composing univariate transformations through additive or multiplicative aggregation. This paper establishes theoretical convergence guarantees for KANs when the univariate components are represented by B-splines. We prove that both additive and hybrid additive-multiplicative KANs attain the minimax-optimal convergence rate $O(n^{-2r/(2r+1)})$ for functions in Sobolev spaces of smoothness $r$. We further derive guidelines for selecting the optimal number of knots in the B-splines. The theory is supported by simulation studies that confirm the predicted convergence rates. These results provide a theoretical foundation for using KANs in nonparametric regression and highlight their potential as a structured alternative to existing methods.

TL;DR: Convergence Rate of Kolmogorov-Arnold Networks