Approximation rates in Besov norms and sample-complexity of Kolmogorov-Arnold networks with residual connections.

Abstract

Inspired by the Kolmogorov-Arnold superposition theorem, Kolmogorov-Arnold Networks (KANs) have recently emerged as an improved backbone for most deep learning frameworks, promising more adaptivity than their multilayer perceptron (MLP) predecessor by allowing for trainable spline-based activation functions. In this paper, we probe the theoretical foundations of the KAN architecture by showing that it can optimally approximate any Besov function in B_p,q^s(X) on a bounded open, or even fractal, domain X in R^d at the optimal approximation rate with respect to any weaker Besov norm B_p,q^α(X); where α < s. We complement our approximation result with a statistical guarantee by bounding the pseudodimension of the relevant class of Res-KANs. As an application of the latter, we directly deduce a dimension-free estimate on the sample complexity of a residual KAN model when learning a function of Besov regularity from N i.i.d. noiseless samples, showing that KANs can learn the smooth maps which they can approximate.