Recently there has been significant theoretical progress on understanding the convergence andgeneralization ofgradient-based methods onnonconvexlosses withoverparameterized models. Nevertheless, manyaspectsofoptimization and generalization and in particular the critical role of small random initialization are not fully understood.

Recently there has been significant theoretical progress on understanding the convergence andgeneralization ofgradient-based methods onnonconvexlosses withoverparameterized models. Nevertheless, manyaspectsofoptimization and generalization and in particular the critical role of small random initialization are not fully understood.

However, since this proof relies on the existence of a convergent subsequence, their proof does not reveal any rate forglobal convergence.

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 $(ε,δ)$-DP and obtain a utility bound of order $\sqrt{d}/(nε)$ (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.