On the Rate of Convergence of Kolmogorov-Arnold Network Regression Estimators

Abstract

Kolmogorov-Arnold Networks (KANs) approximate multivariate functions by composing univariate transformations through additive or multiplicative aggregation. We establish convergence guarantees for KANs whose univariate components are B-splines. The least-squares estimator over the KAN spline sieve attains the rate $O((\log n / n)^{2r/(2r+1)})$, uniformly over a ball of regression functions admitting a KAN representation with univariate components of Sobolev smoothness $r$; a matching lower bound of order $n^{-2r/(2r+1)}$ shows this is minimax optimal up to the logarithmic factor, which we trace to the nonlinearity of the sieve rather than to the architecture. The rate is free of the ambient dimension $d$; this dimension-free exponent reflects the assumed KAN structure of the target, not an escape from the minimax rate $n^{-2r/(2r+d)}$ on Sobolev classes over $[0,1]^d$. We derive a knot-selection rule, show that penalized selection over a dyadic knot grid attains the rate adaptively in the unknown smoothness, and show that univariate components are not identifiable under centering alone, so consistency of the fit does not imply consistency of the components. On targets of exactly known smoothness the fitted risk exponent is at least as steep as the bound in every configuration, and the predicted knot scaling and $k^{-r}$ approximation decay are checked directly.