Optimization, Generalization and Differential Privacy Bounds for Gradient Descent on Kolmogorov-Arnold Networks
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| Format: | Preprint |
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2026
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| author | Wang, Puyu Zhou, Junyu Liznerski, Philipp Kloft, Marius |
| author_facet | Wang, Puyu Zhou, Junyu Liznerski, Philipp Kloft, Marius |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_22409 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimization, Generalization and Differential Privacy Bounds for Gradient Descent on Kolmogorov-Arnold Networks Wang, Puyu Zhou, Junyu Liznerski, Philipp Kloft, Marius Machine Learning Artificial Intelligence 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. |
| title | Optimization, Generalization and Differential Privacy Bounds for Gradient Descent on Kolmogorov-Arnold Networks |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2601.22409 |