Graph convolutional networks (GCN) are viewed as one of the most popular representations among the variants of graph neural networks over graph data and have shown powerful performance in empirical experiments. That $\ell_2$-based graph smoothing enforces the global smoothness of GCN, while (soft) $\ell_1$-based sparse graph learning tends to promote signal sparsity to trade for discontinuity. This paper aims to quantify the trade-off of GCN between smoothness and sparsity, with the help of a general $\ell_p$-regularized $(1<p\leq 2)$ stochastic learning proposed within. While stability-based generalization analyses have been given in prior work for a second derivative objectiveness function, our $\ell_p$-regularized learning scheme does not satisfy such a smooth condition. To tackle this issue, we propose a novel SGD proximal algorithm for GCNs with an inexact operator. For a single-layer GCN, we establish an explicit theoretical understanding of GCN with the $\ell_p$-regularized stochastic learning by analyzing the stability of our SGD proximal algorithm. We conduct multiple empirical experiments to validate our theoretical findings.

翻译：图卷积网络（GCN）被视为图数据上图神经网络变体中最流行的表示之一，并在实证实验中展现出强大的性能。基于$\ell_2$的图平滑强制实现GCN的全局平滑性，而（软）$\ell_1$稀疏图学习倾向于促进信号稀疏性以换取非连续性。本文旨在通过提出的通用$\ell_p$正则化（$1<p\leq 2$）随机学习框架，量化GCN在平滑性与稀疏性之间的权衡。尽管先前的稳定性泛化分析针对二阶导目标函数展开，但我们的$\ell_p$正则化学习方案不满足此类平滑条件。为解决此问题，我们提出一种针对GCN的新型SGD近端算法，并引入非精确算子。针对单层GCN，我们通过分析SGD近端算法的稳定性，建立了$\ell_p$正则化随机学习下GCN的显式理论理解。我们进行了多项实证实验以验证理论发现。