On the Koopman-Based Generalization Bounds for Multi-Task Deep Learning

2025-12-22

AI Summary

Existing generalization bounds for multi-task deep learning are overly loose, particularly those derived from norm-based analyses or Koopman operator theory. Method: This paper proposes a novel analytical framework grounded in Koopman operator theory and Sobolev spaces. It introduces weight matrices with bounded condition numbers and a tailored Sobolev hypothesis space, thereby integrating Koopman operator methods with function-space regularization for the first time. Contribution/Results: The resulting generalization bound is width-independent, scalable, and significantly tighter than conventional norm-based bounds—while remaining applicable even in single-output settings, thus overcoming key modeling limitations of prior Koopman-based bounds. Theoretical analysis demonstrates substantial improvements in both compactness and broad applicability, establishing a new paradigm for generalization analysis in multi-task deep learning.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Computer Vision: Learning & Optimization for CV

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

The paper establishes generalization bounds for multitask deep neural networks using operator-theoretic techniques. The authors propose a tighter bound than those derived from conventional norm based methods by leveraging small condition numbers in the weight matrices and introducing a tailored Sobolev space as an expanded hypothesis space. This enhanced bound remains valid even in single output settings, outperforming existing Koopman based bounds. The resulting framework maintains key advantages such as flexibility and independence from network width, offering a more precise theoretical understanding of multitask deep learning in the context of kernel methods.

Problem

• Establishes generalization bounds for multitask deep neural networks
• Proposes a tighter bound using small condition numbers and Sobolev space
• Offers a precise theoretical understanding of multitask deep learning

Innovation

• Uses operator-theoretic techniques for generalization bounds
• Introduces tailored Sobolev space as expanded hypothesis space
• Leverages small condition numbers for tighter bounds