Existing Graph Convolutional Networks (GCNs) are shallow---the number of the layers is usually not larger than 2. The deeper variants by simply stacking more layers, unfortunately perform worse, even involving well-known tricks like weight penalizing, dropout, and residual connections. This paper reveals that developing deep GCNs mainly encounters two obstacles: and . The over-fitting issue weakens the generalization ability on small graphs, while over-smoothing impedes model training by isolating output representations from the input features with the increase in network depth. Hence, we propose DropEdge, a novel technique to alleviate both issues. At its core, DropEdge randomly removes a certain number of edges from the input graphs, acting like a data augmenter and also a message passing reducer. More importantly, DropEdge enables us to recast a wider range of Convolutional Neural Networks (CNNs) from the image field to the graph domain; in particular, we study DenseNet and InceptionNet in this paper. Extensive experiments on several benchmarks demonstrate that our method allows deep GCNs to achieve promising performance, even when the number of layers exceeds 30---the deepest GCN that has ever been proposed.
Despite the fruitful progress, most previous works only focus on shallow GCNs while the deeper extension is seldom discussed. The work by @cite_30 first introduces the concept of over-smoothing in GCNs, but it never proposes a deep GCN with addressing this issue. Its following study @cite_22 solves over-smoothing by using personalized PageRank that additionally involves the rooted node into the message passing loop; however, the accuracy is still observed to decrease when the depth of GCN increases from 2. The JKNet @cite_10 employs skip connections for multi-hop message passing, and it enables different neighborhood ranges for better structure-aware representation learning. Unexpectedly, as shown in the experiments, the JKNets that obtain the best accuracy have depth less than 3 on all datasets, except the one on Cora where the best result is given by the 6-layer network. In this paper, we propose the notion of DropEdge to overcome both the over-fitting and over-smoothing issues simultaneously, and combine it with various backbone architectures to drive an in-depth analysis on deep GCNs.
@cite_30 @cite_10 @cite_22
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Graph Convolutional Networks (GCNs) have become a crucial tool on learning representations of graph vertices. The main challenge of adapting GCNs on large-scale graphs is the scalability issue that it incurs heavy cost both in computation and memory due to the uncontrollable neighborhood expansion across layers. In this paper, we accelerate the training of GCNs through developing an adaptive layer-wise sampling method. By constructing the network layer by layer in a top-down passway, we sample the lower layer conditioned on the top one, where the sampled neighborhoods are shared by different parent nodes and the over expansion is avoided owing to the fixed-size sampling. More importantly, the proposed sampler is adaptive and applicable for explicit variance reduction, which in turn enhances the training of our method. Furthermore, we propose a novel and economical approach to promote the message passing over distant nodes by applying skip connections. Intensive experiments on several benchmarks verify the effectiveness of our method regarding the classification accuracy while enjoying faster convergence speed. Deep neural network is difficult to train and this predicament becomes worse as the depth increases. The essence of this problem exists in the magnitude of backpropagated errors that will result in gradient vanishing or exploding phenomenon. We show that a variant of regularizer which utilizes orthonormality among different filter banks can alleviate this problem. Moreover, we design a backward error modulation mechanism based on the quasi-isometry assumption between two consecutive parametric layers. Equipped with these two ingredients, we propose several novel optimization solutions that can be utilized for training a specific-structured (repetitively triple modules of Conv-BNReLU) extremely deep convolutional neural network (CNN) WITHOUT any shortcuts identity mappings from scratch. Experiments show that our proposed solutions can achieve distinct improvements for a 44-layer and a 110-layer plain networks on both the CIFAR-10 and ImageNet datasets. Moreover, we can successfully train plain CNNs to match the performance of the residual counterparts. Besides, we propose new principles for designing network structure from the insights evoked by orthonormality. Combined with residual structure, we achieve comparative performance on the ImageNet dataset. Deep neural network is difficult to train and this predicament becomes worse as the depth increases. The essence of this problem exists in the magnitude of backpropagated errors that will result in gradient vanishing or exploding phenomenon. We show that a variant of regularizer which utilizes orthonormality among different filter banks can alleviate this problem. Moreover, we design a backward error modulation mechanism based on the quasi-isometry assumption between two consecutive parametric layers. Equipped with these two ingredients, we propose several novel optimization solutions that can be utilized for training a specific-structured (repetitively triple modules of Conv-BNReLU) extremely deep convolutional neural network (CNN) WITHOUT any shortcuts identity mappings from scratch. Experiments show that our proposed solutions can achieve distinct improvements for a 44-layer and a 110-layer plain networks on both the CIFAR-10 and ImageNet datasets. Moreover, we can successfully train plain CNNs to match the performance of the residual counterparts. Besides, we propose new principles for designing network structure from the insights evoked by orthonormality. Combined with residual structure, we achieve comparative performance on the ImageNet dataset. Many interesting problems in machine learning are being revisited with new deep learning tools. For graph-based semisupervised learning, a recent important development is graph convolutional networks (GCNs), which nicely integrate local vertex features and graph topology in the convolutional layers. Although the GCN model compares favorably with other state-of-the-art methods, its mechanisms are not clear and it still requires a considerable amount of labeled data for validation and model selection. In this paper, we develop deeper insights into the GCN model and address its fundamental limits. First, we show that the graph convolution of the GCN model is actually a special form of Laplacian smoothing, which is the key reason why GCNs work, but it also brings potential concerns of over-smoothing with many convolutional layers. Second, to overcome the limits of the GCN model with shallow architectures, we propose both co-training and self-training approaches to train GCNs. Our approaches significantly improve GCNs in learning with very few labels, and exempt them from requiring additional labels for validation. Extensive experiments on benchmarks have verified our theory and proposals.