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

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

TL;DR

This work extends Koopman-operator-based generalization analysis to multitask deep learning by embedding vector-valued outputs in Sobolev-based vvRKHS spaces and leveraging matrix-valued kernels. It derives multitask Rademacher complexity bounds that couple weight-matrix norms, determinants, and output traces, and accounts for task relations via a joint kernel framework. The results recover tighter bounds than norm-based approaches, with width-independence under orthogonal weights and extensibility to non-injective and convolutional layers. The framework lays the groundwork for future empirical validation and integration with learning-dynamics analyses in multi-task neural networks.

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.

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

TL;DR

This work extends Koopman-operator-based generalization analysis to multitask deep learning by embedding vector-valued outputs in Sobolev-based vvRKHS spaces and leveraging matrix-valued kernels. It derives multitask Rademacher complexity bounds that couple weight-matrix norms, determinants, and output traces, and accounts for task relations via a joint kernel framework. The results recover tighter bounds than norm-based approaches, with width-independence under orthogonal weights and extensibility to non-injective and convolutional layers. The framework lays the groundwork for future empirical validation and integration with learning-dynamics analyses in multi-task neural networks.

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.
Paper Structure (14 sections, 5 theorems, 29 equations, 1 figure)

This paper contains 14 sections, 5 theorems, 29 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related Works
  3. Preliminaries
  4. Notation
  5. Vector-valued Rademacher complexity
  6. Kernel methods for vector-valued responses
  7. Koopman Operator
  8. Vector-Valued Sobolev Space and its Associate vvRKHS
  9. Integral Space of RKHSs
  10. Problem Setup
  11. Koopman-based Bound of Multi-task Rademacher Complexity
  12. Bound for invertible weight matrices
  13. Bound for injective weight matrices
  14. Conclusion

Key Result

Theorem 1

Let $K$ be a $\mathrm{MVK}$. There is a unique Hilbert space $\mathcal{H}_{K} \subset \mathcal{F}(\mathcal{X} , \mathbb{R}^m)$, the $\mathrm{vvRKHS}$ of $K$, such that for all $\mathbf x \in \mathcal{X}$, $\mathbf y \in \mathbb{R}^m$ and $f \in \mathcal{H}_{K}$ we have $\mathbf x' \to K(\mathbf x,\m

Figures (1)

  • Figure 1: Illustration of the proposed network architecture. The network consists of an input layer, one hidden layer, activation function $\sigma_1$, final nonlinear transformation $g$, and an output layer.

Theorems & Definitions (12)

  • Definition 1: (empirical) vector-valued Rademacher complexity
  • Definition 2: Matrix-valued Kernel
  • Theorem 1: wittwar2022, Theorem 2.2.6
  • Definition 3: Sobolev space
  • Definition 4: Vector-Valued Sobolev Space
  • Remark 1
  • Lemma 1
  • Theorem 2
  • proof
  • Corollary 1
  • ...and 2 more