An Anatomy of Graph Neural Networks Going Deep via the Lens of Mutual Information: Exponential Decay vs. Full Preservation

Graph Convolutional Network (GCN) has attracted intensive interests recently. One major limitation of GCN is that it often cannot benefit from using a deep architecture, while traditional CNN and an alternative Graph Neural Network architecture, namely GraphCNN, often achieve better quality with a deeper neural architecture. How can we explain this phenomenon? In this paper, we take the first step towards answering this question. We first conduct a systematic empirical study on the accuracy of GCN, GraphCNN, and ResNet-18 on 2D images and identified relative importance of different factors in architectural design. This inspired a novel theoretical analysis on the mutual information between the input and the output after l GCN/ GraphCNN layers. We identified regimes in which GCN suffers exponentially fast "information lose" and show that GraphCNN requires a much weaker condition for similar behavior to happen. Extending convolutional neural networks (CNN) over images to a graph has attracted intense interest recently. One early attempt is the GCN model proposed by Kipf & Welling (2016a). However, when applying GCN to many practical applications, one discrepancy lingers -- although traditional CNN usually gets higher accuracy when it goes deeper, GCN, as a natural extension of CNN, does not seem to benefit much from going deeper by stacking multiple layers together. This phenomenon has been the focus of multiple recent papers (Li et al., 2018; 2019; Oono & Suzuki, 2019). On the theoretical side, Li et al. (2018) and Oono & Suzuki (2019) identified the problem as oversmoothing -- under certain conditions, when multiple GCN layers are stacked together, the output will converge to a region that is independent of weights and inputs. On the empirical side, Li et al. (2019) showed that many techniques that were designed to train a deep CNN, e.g., the skip connections in ResNet (He et al., 2016a), can also make it easier for GCN to go deeper.