Learning Theory for Kernel Bilevel Optimization

• URL: http://arxiv.org/abs/2502.08457v2
• Date: Tue, 30 Sep 2025 11:58:49 GMT
• Title: Learning Theory for Kernel Bilevel Optimization
• Authors: Fares El Khoury, Edouard Pauwels, Samuel Vaiter, Michael Arbel,
• Abstract summary: We study Kernel Bilevel Optimization (KBO), where the inner objective is optimized over a kernel reproducing space.<n>We derive novel finite-sample generalization bounds for KBO, leveraging tools from empirical process theory.<n>We illustrate our theoretical findings on a synthetic instrumental variable regression task.
• Score: 21.75708217452224
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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