Graph neural networks (GNNs) work remarkably well in semi-supervised node regression, yet a rigorous theory explaining when and why they succeed remains lacking. To address this gap, we study an aggregate-and-readout model that encompasses several common message passing architectures: node features are first propagated over the graph then mapped to responses via a nonlinear function. For least-squares estimation over GNNs with linear graph convolutions and a deep ReLU readout, we prove a sharp non-asymptotic risk bound that separates approximation, stochastic, and optimization errors. The bound makes explicit how performance scales with the fraction of labeled nodes and graph-induced dependence. Approximation guarantees are further derived for graph-smoothing followed by smooth nonlinear readouts, yielding convergence rates that recover classical nonparametric behavior under full supervision while characterizing performance when labels are scarce. Numerical experiments validate our theory, providing a systematic framework for understanding GNN performance and limitations.

翻译：图神经网络（GNNs）在半监督节点回归任务中表现卓越，然而关于其何时及为何能取得成功的严格理论仍显不足。为填补这一空白，我们研究了一个包含多种常见消息传递架构的聚合-读出模型：节点特征首先在图上传导，随后通过非线性函数映射为响应。针对采用线性图卷积与深度ReLU读出层的GNN的最小二乘估计，我们证明了一个精确的非渐近风险界，该界限分离了近似误差、随机误差与优化误差。该界限明确揭示了性能如何随标记节点比例及图结构诱导的依赖性而变化。进一步推导了图平滑后接平滑非线性读出层的近似保证，所得收敛率在完全监督条件下恢复了经典非参数特性，同时刻画了标签稀缺时的性能表现。数值实验验证了我们的理论，为理解GNN的性能与局限提供了系统化框架。