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Generalization Bounds and Model Complexity for Kolmogorov-Arnold Networks

Xianyang Zhang, Huijuan Zhou

TL;DR

This paper analyzes generalization in Kolmogorov-Arnold Networks (KANs), a depth-$L$ architecture where edge-embedded univariate activations replace conventional weights to form a parsimonious, interpretable model for scientific tasks. It develops two theoretical streams: activations as basis-function expansions and activations in a low-rank RKHS (notably Matérn kernels), deriving generalization bounds that depend on per-layer Lipschitz constants and norm constraints while largely avoiding dependence on combinatorial parameters. The results leverage Maurey’s sparsification and covering-number techniques, and are supported by empirical studies showing a strong link between a derived complexity measure and excess risk, suggesting regularization avenues for KAN design and training. Overall, the work provides a principled framework to quantify and control network complexity in KANs, informing activation-function design for improved generalization in science-oriented settings.

Abstract

Kolmogorov-Arnold Network (KAN) is a network structure recently proposed by Liu et al. (2024) that offers improved interpretability and a more parsimonious design in many science-oriented tasks compared to multi-layer perceptrons. This work provides a rigorous theoretical analysis of KAN by establishing generalization bounds for KAN equipped with activation functions that are either represented by linear combinations of basis functions or lying in a low-rank Reproducing Kernel Hilbert Space (RKHS). In the first case, the generalization bound accommodates various choices of basis functions in forming the activation functions in each layer of KAN and is adapted to different operator norms at each layer. For a particular choice of operator norms, the bound scales with the $l_1$ norm of the coefficient matrices and the Lipschitz constants for the activation functions, and it has no dependence on combinatorial parameters (e.g., number of nodes) outside of logarithmic factors. Moreover, our result does not require the boundedness assumption on the loss function and, hence, is applicable to a general class of regression-type loss functions. In the low-rank case, the generalization bound scales polynomially with the underlying ranks as well as the Lipschitz constants of the activation functions in each layer. These bounds are empirically investigated for KANs trained with stochastic gradient descent on simulated and real data sets. The numerical results demonstrate the practical relevance of these bounds.

Generalization Bounds and Model Complexity for Kolmogorov-Arnold Networks

TL;DR

This paper analyzes generalization in Kolmogorov-Arnold Networks (KANs), a depth- architecture where edge-embedded univariate activations replace conventional weights to form a parsimonious, interpretable model for scientific tasks. It develops two theoretical streams: activations as basis-function expansions and activations in a low-rank RKHS (notably Matérn kernels), deriving generalization bounds that depend on per-layer Lipschitz constants and norm constraints while largely avoiding dependence on combinatorial parameters. The results leverage Maurey’s sparsification and covering-number techniques, and are supported by empirical studies showing a strong link between a derived complexity measure and excess risk, suggesting regularization avenues for KAN design and training. Overall, the work provides a principled framework to quantify and control network complexity in KANs, informing activation-function design for improved generalization in science-oriented settings.

Abstract

Kolmogorov-Arnold Network (KAN) is a network structure recently proposed by Liu et al. (2024) that offers improved interpretability and a more parsimonious design in many science-oriented tasks compared to multi-layer perceptrons. This work provides a rigorous theoretical analysis of KAN by establishing generalization bounds for KAN equipped with activation functions that are either represented by linear combinations of basis functions or lying in a low-rank Reproducing Kernel Hilbert Space (RKHS). In the first case, the generalization bound accommodates various choices of basis functions in forming the activation functions in each layer of KAN and is adapted to different operator norms at each layer. For a particular choice of operator norms, the bound scales with the norm of the coefficient matrices and the Lipschitz constants for the activation functions, and it has no dependence on combinatorial parameters (e.g., number of nodes) outside of logarithmic factors. Moreover, our result does not require the boundedness assumption on the loss function and, hence, is applicable to a general class of regression-type loss functions. In the low-rank case, the generalization bound scales polynomially with the underlying ranks as well as the Lipschitz constants of the activation functions in each layer. These bounds are empirically investigated for KANs trained with stochastic gradient descent on simulated and real data sets. The numerical results demonstrate the practical relevance of these bounds.

Paper Structure

This paper contains 16 sections, 16 theorems, 95 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related works
  4. Statistical Analysis
  5. Notation and definitions
  6. Risk analysis: activation functions represented by basis functions
  7. Risk analysis: activation functions in low-rank Sobolev Space
  8. Numerical Studies
  9. Discussions
  10. Appendix
  11. Comparison with MLPs
  12. Improved Bounds Under Sub-exponential tails
  13. A lower bound on the empirical Rademacher complexity
  14. Additional Numerical Results
  15. Auxiliary lemmas
  16. ...and 1 more sections

Key Result

Proposition 1

Suppose $\boldsymbol{\Psi}_i\in\mathcal{F}_i$ is a map from $\mathbb{R}^{d_{i-1}}$ to $\mathbb{R}^{d_i}$, which satisfies $|\boldsymbol{\Psi}({\mathbf{X}})-\boldsymbol{\Psi}({\mathbf{X}}')|_{i}\leq \rho_i |{\mathbf{X}} - {\mathbf{X}}'|_{i-1}$ for $\rho_i > 0$ and ${\mathbf{X}},{\mathbf{X}}'\in\mathb where when $i=0$, the first term in the product is given by $\mathcal{N}\left(\{\boldsymbol{\Psi}({

Figures (4)

  • Figure 1: Illustration of Kolmogorov-Arnold Networks, where the edges are associated with one-dimensional trainable functions while the nodes perform summation.
  • Figure 2: The excess loss and (normalized) complexity of KANs trained with SGD on the four simulated datasets (i--iv) and the MNIST and CIFAR10 datasets. The loss for (iii), (iv), MNIST, and CIFAR10 is the cross-entropy loss, and that for (i) and (ii) is the mean squared error.
  • Figure 3: The training loss, test loss and complexity (normalized with respect to test loss) of KANs trained with SGD on the four simulated datasets (i--iv) and the MNIST and CIFAR10 datasets. The loss for (iii), (iv), MNIST, and CIFAR10 is the cross-entropy loss, and that for (i) and (ii) is the mean squared error.
  • Figure 4: (Top) The excess loss and (normalized) complexity of KANs trained with SGD on the simulated datasets (i) and (iii). The loss for (i) is the mean squared error, and that for (iii) is the cross-entropy loss. The green and red curves in the top-right panel overlap. (Bottom) The ratio of the complexity of the regularized KAN to the non-regularized KAN.

Theorems & Definitions (36)

  • Proposition 1
  • Proposition 2
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • ...and 26 more