Posterior Contraction Rates for Sparse Kolmogorov-Arnold Networks in Anisotropic Besov Spaces

TL;DR

This paper establishes theoretical guarantees for sparse Bayesian Kolmogorov-Arnold networks (KANs), showing they adapt to anisotropic Besov spaces and achieve near-minimax posterior contraction rates, with fixed depth and controlled complexity.

Contribution

It provides the first theoretical analysis of Bayesian KANs, demonstrating their adaptive contraction rates and fixed-depth advantage over traditional sparse MLP models.

Findings

Sparse Bayesian KANs attain near-minimax posterior contraction rates.

The contraction rate depends on the intrinsic anisotropic smoothness.

The approach adapts to unknown smoothness and avoids the curse of dimensionality.

Abstract

We study posterior contraction rates for sparse Bayesian Kolmogorov-Arnold networks (KANs) over anisotropic Besov spaces, providing a statistical foundation of KANs from a Bayesian point of view. We show that sparse Bayesian KANs equipped with spike-and-slab-type sparsity priors attain the near-minimax posterior contraction. In particular, the contraction rate depends on the intrinsic anisotropic smoothness of the underlying function. Moreover, by placing a hyperprior on a single model-size parameter, the resulting posterior adapts to unknown anisotropic smoothness and still achieves the corresponding near-minimax rate. A distinctive feature of our results, compared with those for standard sparse MLP-based models, is that the KAN depth can be kept fixed: owing to the flexibility of learnable spline edge functions, the required approximation complexity is controlled through the network width, spline-grid range and size, and parameter sparsity. Our analysis develops theoretical tools tailored to sparse spline-edge architectures, including approximation and complexity bounds for Bayesian KANs. We then extend to compositional Besov spaces and show that the contraction rates depend on layerwise smoothness and effective dimension of the underlying compositional structure, thereby effectively avoiding the curse of dimensionality. Together, the developed tools and findings advance the theoretical understanding of Bayesian neural networks and provide rigorous statistical foundations for KANs.