Generalization in Graph Neural Networks: Improved PAC-Bayesian Bounds on Graph Diffusion
This work addresses the theoretical understanding of generalization in graph neural networks, which is incremental but provides improved bounds for researchers and practitioners in graph machine learning.
The paper tackled the problem of deriving tighter generalization bounds for graph neural networks by scaling them with the largest singular value of the feature diffusion matrix, resulting in bounds that are numerically much smaller than prior ones for real-world graphs and including a matching lower bound.
Graph neural networks are widely used tools for graph prediction tasks. Motivated by their empirical performance, prior works have developed generalization bounds for graph neural networks, which scale with graph structures in terms of the maximum degree. In this paper, we present generalization bounds that instead scale with the largest singular value of the graph neural network's feature diffusion matrix. These bounds are numerically much smaller than prior bounds for real-world graphs. We also construct a lower bound of the generalization gap that matches our upper bound asymptotically. To achieve these results, we analyze a unified model that includes prior works' settings (i.e., convolutional and message-passing networks) and new settings (i.e., graph isomorphism networks). Our key idea is to measure the stability of graph neural networks against noise perturbations using Hessians. Empirically, we find that Hessian-based measurements correlate with the observed generalization gaps of graph neural networks accurately. Optimizing noise stability properties for fine-tuning pretrained graph neural networks also improves test performance on several graph-level classification tasks.