Graph convolutional networks (GCNs) have achieved promising performance on various graph-based tasks. However they suffer from over-smoothing when stacking more layers. In this paper, we present a quantitative study on this observation and develop novel insights towards the deeper GCN. First, we interpret the current graph convolutional operations from an optimization perspective and argue that over-smoothing is mainly caused by the naive first-order approximation of the solution to the optimization problem. Subsequently, we introduce two metrics to measure the over-smoothing on node-level tasks. Specifically, we calculate the fraction of the pairwise distance between connected and disconnected nodes to the overall distance respectively. Based on our theoretical and empirical analysis, we establish a universal theoretical framework of GCN from an optimization perspective and derive a novel convolutional kernel named GCN+ which has lower parameter amount while relieving the over-smoothing inherently. Extensive experiments on real-world datasets demonstrate the superior performance of GCN+ over state-of-the-art baseline methods on the node classification tasks.

本文从优化的角度解释了当前的图卷积操作存在的过度平滑问题，并引入了两个衡量节点级任务上过度平滑的度量标准，提出了一种新的卷积核GCN+，并在真实的数据集上证明了其在节点分类任务上的优秀性能表现。

从优化角度重新审视半监督节点分类中的图卷积网络