Learning Theory for Kernel Bilevel Optimization
This work addresses a problem relevant to researchers and practitioners in machine learning, particularly those dealing with bilevel optimization and nonparametric settings.
The authors tackled the problem of bilevel optimization in the nonparametric case, specifically Kernel Bilevel Optimization, and derived novel finite-sample generalization bounds. Their results provide a foundation for rigorous theoretical analysis and assessment of statistical accuracy.
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.