Learning Theory for Kernel Bilevel Optimization

TL;DR

This paper establishes a theoretical foundation for kernel-based bilevel optimization in machine learning, deriving generalization bounds and analyzing gradient methods in a nonparametric setting.

Contribution

It introduces the first learning-theoretic analysis of Kernel Bilevel Optimization, providing finite-sample bounds and insights into gradient-based methods in this nonparametric framework.

Findings

Derived novel finite-sample generalization bounds for KBO.

Assessed the statistical accuracy of gradient-based methods in KBO.

Numerical experiments on synthetic data support theoretical results.

Abstract

Bilevel optimization has emerged as a technique for addressing a wide range of machine learning problems that involve an outer objective implicitly determined by the minimizer of an inner problem. While prior works have primarily focused on the parametric setting, a learning-theoretic foundation for bilevel optimization in the nonparametric case remains relatively unexplored. In this paper, we take a first step toward bridging this gap by studying Kernel Bilevel Optimization (KBO), where the inner objective is optimized over a reproducing kernel Hilbert space. This setting enables rich function approximation while providing a foundation for rigorous theoretical analysis. In this context, we derive novel finite-sample generalization bounds for KBO, leveraging tools from empirical process theory. These bounds further allow us to assess the statistical accuracy of gradient-based methods applied to the empirical discretization of KBO. We numerically illustrate our theoretical findings on a synthetic instrumental variable regression task.