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

TL;DR

This paper introduces new operator-theoretic generalization bounds for multi-task deep learning, combining Koopman and Perron Frobenius operators with sketching techniques to improve performance guarantees and address overfitting and underfitting.

Contribution

It develops a novel operator-based framework for generalization bounds in multi-task deep learning, integrating Koopman and Perron Frobenius operators with sketching methods.

Findings

Tighter generalization guarantees than traditional bounds.

Effective sketching techniques for Koopman-based methods.

Performance guarantees for robust and quantile regression applications.

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.