Title: Beyond GNNs: A Sample Efficient Architecture for Graph Problems. Abstract: Despite their popularity in learning problems over graph structured data, existing Graph Neural Networks (GNNs) have inherent limitations for fundamental graph problems such as shortest paths, $k$-connectivity, minimum spanning tree and minimum cuts. In all these instances, it is known that one needs GNNs of high depth, scaling at a polynomial rate with the number of nodes $n$, to provably encode the solution space. This in turn affects their statistical efficiency thus requiring a significant amount of training data in order to obtain networks with good performance. In this work we propose a new hybrid architecture to overcome this limitation. Our proposed architecture that we call as GNNplus networks involve a combination of multiple parallel low depth GNNs along with simple pooling layers involving low depth fully connected networks. We provably demonstrate that for many graph problems, the solution space can be encoded by GNNplus networks using depth that scales only poly-logarithmically in the number of nodes. This significantly improves the amount of training data needed that we establish via improved generalization bounds. Finally, we empirically demonstrate the effectiveness of our proposed architecture for a variety of graph problems.
