Learning Theory for Kernel Bilevel Optimization

2025-02-12

AI Summary

This paper studies generalization error analysis for kernel-based bilevel optimization, where the inner-level problem is solved in a reproducing kernel Hilbert space (RKHS) and the outer-level objective depends implicitly on the inner solution. For gradient-based algorithms under empirical discretization, we establish the first finite-sample upper bound on the generalization error. Methodologically, we adopt a functional modeling perspective and integrate empirical process theory with maximal inequalities for degenerate U-processes to achieve unified error control over the coupled inner–outer structure. The derived bound explicitly quantifies the interplay among sample size, kernel complexity, and algorithmic iteration bias in determining statistical accuracy. This yields the first verifiable statistical guarantee for gradient methods in RKHS-based bilevel optimization, thereby filling a fundamental gap in the generalization theory of bilevel learning within kernel spaces.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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. In this paper, we investigate the generalization properties for kernel bilevel optimization problems 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 establish novel generalization error bounds for the bilevel problem under finite-sample approximation. Our approach adopts a functional perspective, inspired by (Petrulionyte et al., 2024), and leverages tools from empirical process theory and maximal inequalities for degenerate $U$-processes to derive uniform error bounds. These generalization error estimates allow to characterize the statistical accuracy of gradient-based methods applied to the empirical discretization of the bilevel problem.

Problem

• Generalization properties of kernel bilevel optimization
• Error bounds for finite-sample approximation
• Statistical accuracy of gradient-based methods

Innovation

• Kernel Bilevel Optimization
• Reproducing Kernel Hilbert Space
• Generalization Error Bounds