From Graph Low-Rank Global Attention to 2-FWL Approximation

Graph Neural Networks (GNNs) are known to have an expressive power bounded by that of the vertex coloring algorithm (Xu et al., 2019a; Morris et al., 2018). However, for rich node features, such a bound does not exist and GNNs can be shown to be universal, namely, have the theoretical ability to approximate arbitrary graph functions. It is well known, however, that expressive power alone does not imply good generalization. In an effort to improve generalization of GNNs we suggest the Low-Rank Global Attention (LRGA) module, taking advantage of the efficiency of low rank matrix-vector multiplication, that improves the algorithmic alignment (Xu et al., 2019b) of GNNs with the 2-folklore Weisfeiler-Lehman (FWL) algorithm; 2-FWL is a graph isomorphism algorithm that is strictly more powerful than vertex coloring. Concretely, we: (i) formulate 2-FWL using polynomial kernels; (ii) show LRGA aligns with this 2-FWL formulation; and (iii) bound the sample complexity of the kernel's feature map when learned with a randomly initialized two-layer MLP. The latter means the generalization error can be made arbitrarily small when training LRGA to learn the 2-FWL algorithm. From a practical point of view, augmenting existing GNN layers with LRGA produces state of the art results on most datasets in a GNN standard benchmark.