通过李普希茨损失函数提升几何图神经网络泛化能力
Generalization of Geometric Graph Neural Networks with Lipschitz Loss Functions
- 基于李普希茨损失推导图神经网络泛化误差上界
- 采样点越多、流形维度越低，泛化性能越强
- 单一大图即可实现跨图泛化，突破传统规模限制
本文研究几何图神经网络（GNN）的泛化能力。考虑在嵌入流形上随机采样点构建的几何图，其拓扑信息被保留。我们证明了该GNN最优经验风险与最优统计风险之间的泛化差距，该差距随流形采样点数量增加而减小，随底层流形维度增加而增大。这一泛化差距表明，基于采样点构建的图训练的GNN可推广至同一流形上的其他未见图。关键发现是：泛化能力可通过单一大图实现，不再受限于图的规模。该结果基于非渐近收敛分析，即GNN在采样图上的表现趋近于底层流形神经网络（MNN）。实验在多个真实数据集上验证了理论结论。
原文摘要 · Abstract (English)
In this paper, we study the generalization capabilities of geometric graph neural networks (GNNs). We consider GNNs over a geometric graph constructed from a finite set of randomly sampled points over an embedded manifold with topological information captured. We prove a generalization gap between the optimal empirical risk and the optimal statistical risk of this GNN, which decreases with the number of sampled points from the manifold and increases with the dimension of the underlying manifold. This generalization gap ensures that the GNN trained on a graph on a set of sampled points can be utilized to process other unseen graphs constructed from the same underlying manifold. The most important observation is that the generalization capability can be realized with one large graph instead of being limited to the size of the graph as in previous results. The generalization gap is derived based on the non-asymptotic convergence result of a GNN on the sampled graph to the underlying manifold neural networks (MNNs). We verify this theoretical result with experiments on multiple real-world datasets.
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