Optimization, Generalization and Differential Privacy Bounds for Gradient Descent on Kolmogorov-Arnold Networks

TL;DR

This paper provides a theoretical analysis of training dynamics, generalization, and differential privacy bounds for Kolmogorov--Arnold Networks (KANs), highlighting the importance of network width and privacy constraints.

Contribution

It introduces the first principled bounds for KAN training, generalization, and privacy, revealing the necessity of polylogarithmic width under differential privacy.

Findings

Polylogarithmic width suffices for optimization and generalization in KANs.

Differential privacy requires noise proportional to input dimension and sample size.

Polylogarithmic width is necessary for privacy, indicating a gap from non-private training.

Abstract

Kolmogorov--Arnold Networks (KANs) have recently emerged as a structured alternative to standard MLPs, yet a principled theory for their training dynamics, generalization, and privacy properties remains limited. In this paper, we analyze gradient descent (GD) for training two-layer KANs and derive general bounds that characterize their training dynamics, generalization, and utility under differential privacy (DP). As a concrete instantiation, we specialize our analysis to logistic loss under an NTK-separable assumption, where we show that polylogarithmic network width suffices for GD to achieve an optimization rate of order $1/T$ and a generalization rate of order $1/n$, with $T$ denoting the number of GD iterations and $n$ the sample size. In the private setting, we characterize the noise required for $(\epsilon,\delta)$-DP and obtain a utility bound of order $\sqrt{d}/(n\epsilon)$ (with $d$ the input dimension), matching the classical lower bound for general convex Lipschitz problems. Our results imply that polylogarithmic width is not only sufficient but also necessary under differential privacy, revealing a qualitative gap between non-private (sufficiency only) and private (necessity also emerges) training regimes. Experiments further illustrate how these theoretical insights can guide practical choices, including network width selection and early stopping.