Operator-Based Generalization Bound for Deep Learning: Insights on Multi-Task Learning

2025-12-22

AI Summary

This paper addresses the overly loose and computationally expensive generalization bounds of deep neural networks and deep kernel methods in multi-task learning. To tackle this, we propose a novel operator-theoretic framework grounded in the Koopman and Perron–Frobenius (PF) operators. We introduce, for the first time, a vector-valued deep reproducing kernel Hilbert space (RKHS) architecture tailored to multi-task learning, where kernel refinement explicitly captures the interplay between underfitting and overfitting mechanisms. By integrating randomized sketching, Rademacher complexity analysis, and Lipschitz-loss-based reasoning, we derive tight, computationally tractable generalization bounds with provable excess risk guarantees. The framework supports practical learning tasks—including robust regression and quantile regression—while establishing a new theoretical foundation for generalization in multi-task deep learning.

Technology Category

• Machine Learning: Kernel Methods
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Learning to Search

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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

This paper presents novel generalization bounds for vector-valued neural networks and deep kernel methods, focusing on multi-task learning through an operator-theoretic framework. Our key development lies in strategically combining a Koopman based approach with existing techniques, achieving tighter generalization guarantees compared to traditional norm-based bounds. To mitigate computational challenges associated with Koopman-based methods, we introduce sketching techniques applicable to vector valued neural networks. These techniques yield excess risk bounds under generic Lipschitz losses, providing performance guarantees for applications including robust and multiple quantile regression. Furthermore, we propose a novel deep learning framework, deep vector-valued reproducing kernel Hilbert spaces (vvRKHS), leveraging Perron Frobenius (PF) operators to enhance deep kernel methods. We derive a new Rademacher generalization bound for this framework, explicitly addressing underfitting and overfitting through kernel refinement strategies. This work offers novel insights into the generalization properties of multitask learning with deep learning architectures, an area that has been relatively unexplored until recent developments.

Problem

• Develops generalization bounds for vector-valued neural networks and deep kernel methods
• Introduces sketching techniques to reduce computational complexity in Koopman-based approaches
• Proposes a deep vvRKHS framework to address underfitting and overfitting in multitask learning

Innovation

• Koopman approach combined with sketching techniques
• Deep vector-valued RKHS with PF operators
• Rademacher bound addressing underfitting and overfitting