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Register a new userStability and Generalization of Graph Convolutional Neural Networks
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Added by
Saurabh Verma
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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Inspired by convolutional neural networks on 1D and 2D data, graph convolutional neural networks (GCNNs) have been developed for various learning tasks on graph data, and have shown superior performance on real-world datasets. Despite their success, there is a dearth of theoretical explorations of GCNN models such as their generalization properties. In this paper, we take a first step towards developing a deeper theoretical understanding of GCNN models by analyzing the stability of single-layer GCNN models and deriving their generalization guarantees in a semi-supervised graph learning setting. In particular, we show that the algorithmic stability of a GCNN model depends upon the largest absolute eigenvalue of its graph convolution filter. Moreover, to ensure the uniform stability needed to provide strong generalization guarantees, the largest absolute eigenvalue must be independent of the graph size. Our results shed new insights on the design of new & improved graph convolution filters with guaranteed algorithmic stability. We evaluate the generalization gap and stability on various real-world graph datasets and show that the empirical results indeed support our theoretical findings. To the best of our knowledge, we are the first to study stability bounds on graph learning in a semi-supervised setting and derive generalization bounds for GCNN models.
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Graph convolutional neural networks (GCNNs) are nonlinear processing tools to learn representations from network data. A key property of GCNNs is their stability to graph perturbations. Current analysis considers deterministic perturbations but fails to provide relevant insights when topological changes are random. This paper investigates the stability of GCNNs to stochastic graph perturbations induced by link losses. In particular, it proves the expected output difference between the GCNN over random perturbed graphs and the GCNN over the nominal graph is upper bounded by a factor that is linear in the link loss probability. We perform the stability analysis in the graph spectral domain such that the result holds uniformly for any graph. This result also shows the role of the nonlinearity and the architecture width and depth, and allows identifying handle to improve the GCNN robustness. Numerical simulations on source localization and robot swarm control corroborate our theoretical findings.
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Graph convolutional neural networks (GCNs) embed nodes in a graph into Euclidean space, which has been shown to incur a large distortion when embedding real-world graphs with scale-free or hierarchical structure. Hyperbolic geometry offers an exciting alternative, as it enables embeddings with much smaller distortion. However, extending GCNs to hyperbolic geometry presents several unique challenges because it is not clear how to define neural network operations, such as feature transformation and aggregation, in hyperbolic space. Furthermore, since input features are often Euclidean, it is unclear how to transform the features into hyperbolic embeddings with the right amount of curvature. Here we propose Hyperbolic Graph Convolutional Neural Network (HGCN), the first inductive hyperbolic GCN that leverages both the expressiveness of GCNs and hyperbolic geometry to learn inductive node representations for hierarchical and scale-free graphs. We derive GCN operations in the hyperboloid model of hyperbolic space and map Euclidean input features to embeddings in hyperbolic spaces with different trainable curvature at each layer. Experiments demonstrate that HGCN learns embeddings that preserve hierarchical structure, and leads to improved performance when compared to Euclidean analogs, even with very low dimensional embeddings: compared to state-of-the-art GCNs, HGCN achieves an error reduction of up to 63.1% in ROC AUC for link prediction and of up to 47.5% in F1 score for node classification, also improving state-of-the art on the Pubmed dataset.
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