Diagonal Over-parameterization in Reproducing Kernel Hilbert Spaces as an Adaptive Feature Model: Generalization and Adaptivity

2025-01-15

AI Summary

Conventional kernel methods in reproducing kernel Hilbert spaces (RKHS) suffer from limited adaptability to target function structures due to fixed kernel functions. Method: We propose a diagonal adaptive kernel model that jointly learns the diagonal eigenvalues of the kernel operator and output coefficients during training. This is the first incorporation of diagonal overparameterization into RKHS, revealing eigenvalue dynamics as an implicit feature learning mechanism; we further establish theoretically that added depth enhances generalization and adaptability via implicit regularization. Results: Experiments demonstrate substantial improvements in generalization, especially under kernel–target mismatch. We prove theoretically that adaptability arises from end-to-end optimization of eigenvalues—providing a novel perspective on how deep kernel methods and neural networks surpass classical kernel-based approaches in structural adaptivity.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Learning to Search
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

This paper introduces a diagonal adaptive kernel model that dynamically learns kernel eigenvalues and output coefficients simultaneously during training. Unlike fixed-kernel methods tied to the neural tangent kernel theory, the diagonal adaptive kernel model adapts to the structure of the truth function, significantly improving generalization over fixed-kernel methods, especially when the initial kernel is misaligned with the target. Moreover, we show that the adaptivity comes from learning the right eigenvalues during training, showing a feature learning behavior. By extending to deeper parameterization, we further show how extra depth enhances adaptability and generalization. This study combines the insights from feature learning and implicit regularization and provides new perspective into the adaptivity and generalization potential of neural networks beyond the kernel regime.

Problem

• Adaptive Kernel Models
• Diagonal Adaptation
• Self-learning Capability

Innovation

• Diagonal Adaptive Kernel Model
• Self-adjusting Kernel Eigenvalues
• Enhanced Generalization and Adaptability