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

TL;DR

This paper develops improved theoretical generalization bounds for multitask deep neural networks using Koopman operator techniques, emphasizing tighter bounds through condition numbers and Sobolev spaces, applicable even in single output scenarios.

Contribution

It introduces a novel, tighter generalization bound for multitask deep learning based on operator theory, surpassing previous Koopman-based bounds and applicable to single output models.

Findings

Tighter generalization bounds than previous methods.

Bounds remain valid in single output settings.

Framework is flexible and independent of network width.

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.