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Optimal Abstractions for Verifying Properties of Kolmogorov-Arnold Networks (KANs)

Noah Schwartz, Chandra Kanth Nagesh, Sriram Sankaranarayanan, Ramneet Kaur, Tuhin Sahai, Susmit Jha

TL;DR

A novel approach for verifying properties of Kolmogorov-Arnold Networks, a class of neural networks characterized by nonlinear, univariate activation functions typically implemented as piecewise polynomial splines or Gaussian processes, by replacing each KAN unit with a piecewise affine function.

Abstract

We present a novel approach for verifying properties of Kolmogorov-Arnold Networks (KANs), a class of neural networks characterized by nonlinear, univariate activation functions typically implemented as piecewise polynomial splines or Gaussian processes. Our method creates mathematical ``abstractions'' by replacing each KAN unit with a piecewise affine (PWA) function, providing both local and global error estimates between the original network and its approximation. These abstractions enable property verification by encoding the problem as a Mixed Integer Linear Program (MILP), determining whether outputs satisfy specified properties when inputs belong to a given set. A critical challenge lies in balancing the number of pieces in the PWA approximation: too many pieces add binary variables that make verification computationally intractable, while too few pieces create excessive error margins that yield uninformative bounds. Our key contribution is a systematic framework that exploits KAN structure to find optimal abstractions. By combining dynamic programming at the unit level with a knapsack optimization across the network, we minimize the total number of pieces while guaranteeing specified error bounds. This approach determines the optimal approximation strategy for each unit while maintaining overall accuracy requirements. Empirical evaluation across multiple KAN benchmarks demonstrates that the upfront analysis costs of our method are justified by superior verification results.

Optimal Abstractions for Verifying Properties of Kolmogorov-Arnold Networks (KANs)

TL;DR

A novel approach for verifying properties of Kolmogorov-Arnold Networks, a class of neural networks characterized by nonlinear, univariate activation functions typically implemented as piecewise polynomial splines or Gaussian processes, by replacing each KAN unit with a piecewise affine function.

Abstract

We present a novel approach for verifying properties of Kolmogorov-Arnold Networks (KANs), a class of neural networks characterized by nonlinear, univariate activation functions typically implemented as piecewise polynomial splines or Gaussian processes. Our method creates mathematical ``abstractions'' by replacing each KAN unit with a piecewise affine (PWA) function, providing both local and global error estimates between the original network and its approximation. These abstractions enable property verification by encoding the problem as a Mixed Integer Linear Program (MILP), determining whether outputs satisfy specified properties when inputs belong to a given set. A critical challenge lies in balancing the number of pieces in the PWA approximation: too many pieces add binary variables that make verification computationally intractable, while too few pieces create excessive error margins that yield uninformative bounds. Our key contribution is a systematic framework that exploits KAN structure to find optimal abstractions. By combining dynamic programming at the unit level with a knapsack optimization across the network, we minimize the total number of pieces while guaranteeing specified error bounds. This approach determines the optimal approximation strategy for each unit while maintaining overall accuracy requirements. Empirical evaluation across multiple KAN benchmarks demonstrates that the upfront analysis costs of our method are justified by superior verification results.
Paper Structure (22 sections, 9 theorems, 26 equations, 3 figures, 3 tables)

This paper contains 22 sections, 9 theorems, 26 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Verification Problems and Their Significance
  3. Related Work
  4. Preliminaries
  5. Range Analysis Problem:
  6. Optimal Piecewise Affine (PWA) Approximations for Univariate Functions
  7. Discretization vs. Continuous Error:
  8. Trade-Off Table:
  9. Optimal PWA for Entire KAN
  10. Optimal KAN Abstraction as Knapsack
  11. Mixed Integer Linear Programming Encoding
  12. Experiments
  13. Setup
  14. Evaluation Tasks and Results
  15. Sensitivity analysis on application driven tasks
  16. ...and 7 more sections

Key Result

Lemma 1

For each unit $\psi^{(i)}_{j,k}$, there exists a constant $\mathsf{L}^{(i)}_{j,k} \geq 0$ such that all $z, \hat{z}$,

Figures (3)

  • Figure 1: An illustration of the key steps of our approach
  • Figure 2: A two-layer KAN with three inputs $\mathbf{x} = (x_1, x_2, x_3)$ and three outputs $\mathbf{y} = (y_1, y_2, y_3)$, i.e., $K=2$ with $n_1 = 3$, $n_2 = 2$, $n_3 = 3$. Each layer is shaded differently.
  • Figure 3: Input perturbation radius versus average output bound width on four tasks for KAN Optimized and Empirical bound widths.

Theorems & Definitions (15)

  • Definition 1: Kolmogorov-Arnold Networks (KAN)
  • Lemma 1
  • proof
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 2
  • proof
  • Theorem 4
  • ...and 5 more