A Non-Asymptotic Analysis of Oversmoothing in Graph Neural Networks
This addresses a central challenge in building deeper GNNs for graph-based learning tasks, providing a non-asymptotic analysis that clarifies the oversmoothing mechanism, though it is incremental as it builds on prior asymptotic studies.
The paper tackles the oversmoothing problem in Graph Neural Networks by distinguishing between mixing and denoising effects in graph convolutions, showing that oversmoothing occurs when mixing dominates denoising, with a transition at O(log N/log(log N)) layers for dense graphs, and finds that Personalized PageRank mitigates but does not eliminate oversmoothing, with shallow depths often performing best.
Oversmoothing is a central challenge of building more powerful Graph Neural Networks (GNNs). While previous works have only demonstrated that oversmoothing is inevitable when the number of graph convolutions tends to infinity, in this paper, we precisely characterize the mechanism behind the phenomenon via a non-asymptotic analysis. Specifically, we distinguish between two different effects when applying graph convolutions -- an undesirable mixing effect that homogenizes node representations in different classes, and a desirable denoising effect that homogenizes node representations in the same class. By quantifying these two effects on random graphs sampled from the Contextual Stochastic Block Model (CSBM), we show that oversmoothing happens once the mixing effect starts to dominate the denoising effect, and the number of layers required for this transition is $O(\log N/\log (\log N))$ for sufficiently dense graphs with $N$ nodes. We also extend our analysis to study the effects of Personalized PageRank (PPR), or equivalently, the effects of initial residual connections on oversmoothing. Our results suggest that while PPR mitigates oversmoothing at deeper layers, PPR-based architectures still achieve their best performance at a shallow depth and are outperformed by the graph convolution approach on certain graphs. Finally, we support our theoretical results with numerical experiments, which further suggest that the oversmoothing phenomenon observed in practice can be magnified by the difficulty of optimizing deep GNN models.