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

Mahdi Mohammadigohari, Giuseppe Di Fatta, Giuseppe Nicosia, Panos M. Pardalos

TL;DR

The paper develops an operator-theoretic framework for analyzing generalization in vector-valued, multi-task deep learning and deep kernel methods. By integrating Koopman operator-based bounds with vector-valued reproducing kernel Hilbert space (vvRKHS) theory and introducing Perron–Frobenius (PF) operators for deep vvRKHS, it derives tighter Rademacher-based generalization bounds and layer-wise interpretations. It also proposes practical advances, including input sketching for vector-valued nets and a PF-guided deep vvRKHS architecture, with bounds addressing underfitting and overfitting through kernel refinement. The work advances theoretical understanding of multitask learning with deep networks and deep kernels, offering pathways for regularization and scalable analysis. Future directions include empirical validation on Lipschitz losses and extending the PF framework to broader kernel classes.

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.

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

TL;DR

The paper develops an operator-theoretic framework for analyzing generalization in vector-valued, multi-task deep learning and deep kernel methods. By integrating Koopman operator-based bounds with vector-valued reproducing kernel Hilbert space (vvRKHS) theory and introducing Perron–Frobenius (PF) operators for deep vvRKHS, it derives tighter Rademacher-based generalization bounds and layer-wise interpretations. It also proposes practical advances, including input sketching for vector-valued nets and a PF-guided deep vvRKHS architecture, with bounds addressing underfitting and overfitting through kernel refinement. The work advances theoretical understanding of multitask learning with deep networks and deep kernels, offering pathways for regularization and scalable analysis. Future directions include empirical validation on Lipschitz losses and extending the PF framework to broader kernel classes.

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.
Paper Structure (15 sections, 4 theorems, 25 equations, 2 figures)

This paper contains 15 sections, 4 theorems, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related works
  3. Mathematical Preliminaries
  4. Key notations and definitions
  5. Vector-valued Sobolev spaces
  6. Vector-valued Rademacher complexity
  7. Operator Representation framework for deep learning
  8. Generalization bounds via Rademacher complexity
  9. Upper bound for injective weight matrices
  10. Integrating the Koopman-based bound
  11. Excess risk of p-sparsified sketched estimator
  12. Operator Representation framework for deep vvRKHS
  13. Deep vvRKHS
  14. PF generalization bound for deep vvRKHS
  15. Conclusion

Key Result

Lemma 1

The Rademacher complexity $\widehat{\mathfrak{R}}_n^m(\mathcal{F}_{ \mathrm{inj}})$ is bounded as

Figures (2)

  • Figure 1: Illustration of the proposed network architecture (adapted from Mohammadigohari2025koopman, Figure 1). The network consists of an input layer, one hidden layer, activation function $\sigma_1$, final nonlinear transformation $g$, and an output layer.
  • Figure 2: Illustration of the proposed deep vvRKHS (adapted from hashimoto2023deep, Figure 1). For the autoencoder, the encoder is represented by $f_1 \circ f_2$, and the decoder by $f_3 \circ f_4$.

Theorems & Definitions (11)

  • Definition 1: (empirical) vector-valued Rademacher complexity
  • Lemma 1
  • Theorem 1
  • proof
  • Remark 1
  • Definition 2
  • Corollary 1
  • Proposition 1
  • proof
  • Remark 2
  • ...and 1 more