Generalization Bounds for Message Passing Networks on Mixture of Graphons (2404.03473v1)

Abstract: We study the generalization capabilities of Message Passing Neural Networks (MPNNs), a prevalent class of Graph Neural Networks (GNN). We derive generalization bounds specifically for MPNNs with normalized sum aggregation and mean aggregation. Our analysis is based on a data generation model incorporating a finite set of template graphons. Each graph within this framework is generated by sampling from one of the graphons with a certain degree of perturbation. In particular, we extend previous MPNN generalization results to a more realistic setting, which includes the following modifications: 1) we analyze simple random graphs with Bernoulli-distributed edges instead of weighted graphs; 2) we sample both graphs and graph signals from perturbed graphons instead of clean graphons; and 3) we analyze sparse graphs instead of dense graphs. In this more realistic and challenging scenario, we provide a generalization bound that decreases as the average number of nodes in the graphs increases. Our results imply that MPNNs with higher complexity than the size of the training set can still generalize effectively, as long as the graphs are sufficiently large.

• The paper establishes uniform generalization bounds showing that MPNN error decreases with increasing average graph size.
• It extends analysis to sparse and noisy graph-signals from perturbed graphons, enhancing applicability in realistic settings.
• The study highlights the trade-off between graph sparsity and generalization, providing guidance for robust MPNN design.

Generalization Bounds for Message Passing Networks on Mixtures of Graphons

Introduction

Graph Neural Networks (GNNs), particularly Message Passing Neural Networks (MPNNs), have emerged as powerful tools for learning on graph-structured data. Their success in learning meaningful representations from such data has seen widespread application across various domains, including bioinformatics, social network analysis, and computational chemistry. Despite their practical effectiveness, a comprehensive understanding of the theoretical properties that govern the generalization abilities of MPNNs remains an area of active research. In this context, we analyze how MPNNs generalize in supervised learning tasks, especially in settings that involve sparse and noisy graph-signals sampled from a finite set of graphons. Our work extends previous findings to more realistic scenarios, providing insights into the generalization behavior of MPNNs when applied to complex graph-structured data.

Uniform Generalization Bounds

Generalization bounds provide a measure to gauge how well a model learned on a finite set of samples can perform on unseen data drawn from the same distribution. For MPNNs operating on graph-signals, we derive generalization bounds under the assumption that these signals are sampled from a finite set of perturbed graphons, which can better represent the nature of real-world graphs. Unlike previous studies that mainly focused on dense graphs, our analysis encompasses sparse, simple random noisy graph-signals. This extension is crucial for understanding MPNN generalization in more varied and realistic contexts.

Our methodology revolves around quantifying the representativeness error, which measures the discrepancy between the empirical risk (loss on training data) and the expected loss (over the data distribution). We establish that, for MPNNs, this error decreases with the increasing average number of nodes in the graphs, under certain regularity conditions of the graphons and signals. Importantly, our bounds remain valid even when the complexity of the MPNN exceeds the size of the training set, provided that the graphs are sufficiently large.

Key Results and Implications

The main theoretical contribution of our work is the formulation of uniform generalization bounds for MPNNs that are applicable to a more generalized setting. We demonstrate that these bounds are tight in specific scenarios, showing an inverse relationship between the generalization error and the average graph size. This finding is particularly relevant for practical applications, suggesting that MPNN models can generalize well even for complex networks, as long as the underlying graphs in the dataset have a reasonably large size.

Furthermore, our analysis sheds light on the impact of graph sparsity and noise on generalization. We find that increasing the sparsity of the graphs (controlled by the parameter $\alpha$) tends to slow down the rate at which the generalization error decreases. However, provided that $\alpha < \frac{1}{2(D_\chi+2)}$, where $D_\chi$ is the Minkowski dimension of the graphon domain, the generalization error still converges to zero. This result highlights the trade-off between representational capacity (affected by sparsity) and generalization.

Future Directions

While our work significantly advances the theoretical understanding of MPNN generalization, several avenues remain open for further research. One such direction is extending our analysis to other types of aggregations beyond mean and normalized sum, such as max aggregation, which could provide a comprehensive understanding of MPNN behavior across different architectures. Additionally, investigating how our theoretical findings translate to practical scenarios involving real-world datasets could yield valuable insights into designing more robust and generalizable MPNNs.

In conclusion, our paper provides a novel theoretical framework for understanding the generalization capabilities of MPNNs in handling sparse, noisy graph-signals. By extending the analysis to more realistic settings, we offer valuable guidance for applying MPNNs effectively across a wide range of applications.