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

TL;DR

This paper introduces a systematic framework for creating optimal piecewise affine abstractions of Kolmogorov-Arnold Networks, enabling efficient property verification with guaranteed error bounds and minimized complexity.

Contribution

It develops a novel method combining dynamic programming and knapsack optimization to find optimal abstractions for KANs, balancing accuracy and computational tractability.

Findings

Outperforms existing verification methods on KAN benchmarks.

Provides tight error bounds with fewer pieces in the PWA approximation.

Reduces verification computational cost through optimal abstraction strategy.

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.