Kolmogorov-Arnold Networks: Approximation and Learning Guarantees for Functions and their Derivatives

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 perception (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 $B^{s}_{p,q}(\mathcal{X})$ on a bounded open, or even fractal, domain $\mathcal{X}$ in $\mathbb{R}^d$ at the optimal approximation rate with respect to any weaker Besov norm $B^{\alpha}_{p,q}(\mathcal{X})$; where $\alpha < s$. We complement our approximation guarantee with 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. Our KAN architecture incorporates contemporary deep learning wisdom by leveraging residual/skip connections between layers.