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Towards Bridging Generalization and Expressivity of Graph Neural Networks

Shouheng Li, Floris Geerts, Dongwoo Kim, Qing Wang

TL;DR

This work addresses the intricate relationship between expressivity and generalization in GNNs by introducing a novel framework that connects GNN generalization to the variance in graph structures they can capture, and proposes a $k-variance margin-based generalization bound that characterizes the structural properties of graph embeddings in terms of their upper-bounded expressive power.

Abstract

Expressivity and generalization are two critical aspects of graph neural networks (GNNs). While significant progress has been made in studying the expressivity of GNNs, much less is known about their generalization capabilities, particularly when dealing with the inherent complexity of graph-structured data. In this work, we address the intricate relationship between expressivity and generalization in GNNs. Theoretical studies conjecture a trade-off between the two: highly expressive models risk overfitting, while those focused on generalization may sacrifice expressivity. However, empirical evidence often contradicts this assumption, with expressive GNNs frequently demonstrating strong generalization. We explore this contradiction by introducing a novel framework that connects GNN generalization to the variance in graph structures they can capture. This leads us to propose a $k$-variance margin-based generalization bound that characterizes the structural properties of graph embeddings in terms of their upper-bounded expressive power. Our analysis does not rely on specific GNN architectures, making it broadly applicable across GNN models. We further uncover a trade-off between intra-class concentration and inter-class separation, both of which are crucial for effective generalization. Through case studies and experiments on real-world datasets, we demonstrate that our theoretical findings align with empirical results, offering a deeper understanding of how expressivity can enhance GNN generalization.

Towards Bridging Generalization and Expressivity of Graph Neural Networks

TL;DR

This work addresses the intricate relationship between expressivity and generalization in GNNs by introducing a novel framework that connects GNN generalization to the variance in graph structures they can capture, and proposes a $k-variance margin-based generalization bound that characterizes the structural properties of graph embeddings in terms of their upper-bounded expressive power.

Abstract

Expressivity and generalization are two critical aspects of graph neural networks (GNNs). While significant progress has been made in studying the expressivity of GNNs, much less is known about their generalization capabilities, particularly when dealing with the inherent complexity of graph-structured data. In this work, we address the intricate relationship between expressivity and generalization in GNNs. Theoretical studies conjecture a trade-off between the two: highly expressive models risk overfitting, while those focused on generalization may sacrifice expressivity. However, empirical evidence often contradicts this assumption, with expressive GNNs frequently demonstrating strong generalization. We explore this contradiction by introducing a novel framework that connects GNN generalization to the variance in graph structures they can capture. This leads us to propose a -variance margin-based generalization bound that characterizes the structural properties of graph embeddings in terms of their upper-bounded expressive power. Our analysis does not rely on specific GNN architectures, making it broadly applicable across GNN models. We further uncover a trade-off between intra-class concentration and inter-class separation, both of which are crucial for effective generalization. Through case studies and experiments on real-world datasets, we demonstrate that our theoretical findings align with empirical results, offering a deeper understanding of how expressivity can enhance GNN generalization.

Paper Structure

This paper contains 19 sections, 11 theorems, 25 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Preliminaries
  4. Graphs and homomorphisms.
  5. Graph neural networks and WL.
  6. Graph encoders.
  7. Wasserstein distance.
  8. Graph Encoders: Key Properties
  9. Generalization Analysis
  10. Generalization bound
  11. Concentration and separation
  12. Remarks.
  13. Case Studies
  14. Experiments
  15. Results and discussion
  16. ...and 4 more sections

Key Result

Lemma 4.1

Let $\phi: \mathcal{G} \to \mathcal{Z}_\phi$ and $\phi': \mathcal{G} \to \mathcal{Z}_{\phi'}$ be two graph encoders such that $\phi \sqsubseteq \phi'$ holds. Then there exists a function $f: \mathcal{Z}_\phi \to \mathcal{Z}_{\phi'}$ such that $\phi' = f \circ \phi$.

Figures (2)

  • Figure 1: Loss gaps and bounds
  • Figure 2: Bound of MPNN and $C_3$-MPNN

Theorems & Definitions (15)

  • Definition 4.1
  • Lemma 4.1
  • Proposition 4.1
  • Proposition 4.1
  • Corollary 4.2
  • Theorem 5.1
  • Proposition 5.1
  • Lemma B.0
  • proof
  • Proposition B.0
  • ...and 5 more