Energy Guided Smoothness to Improve Robustness

ICLR 2026 Conference Submission20113 Authors

Keywords: Graph neural networks, Dirichlet Energy, Smoothness Bias, Graph Classification, Robustness, Label noise, Classification

TL;DR: Robustness of GNNs to label noise is governed by the smoothness of their representations, quantified via Dirichlet energy, and we proposed three strategies to preserve this smoothness.

Abstract: Graph Neural Networks (GNNs) perform well on graph classification tasks but are notably susceptible to label noise, leading to compromised generalization and overfitting. We investigate GNNs' robustness, identify generalization failure modes and causes, and prove our hypotheses with three robust GNN training methods. Specifically, GNN generalization is compromised by label noise in simpler tasks (few classes), low-order graphs (few nodes), or highly parameterized models. Focusing on graph classification, we show the link between GNN robustness and the smoothness of learned node representations, as quantified by the Dirichlet energy. We show that GNN learns smoother representations with decreasing Dirichlet energy during training, until the model fits on noisy labels, adding high-frequency components to the representations. To verify our analysis, we propose three robust training strategies for GNNs: (a) a spectral inductive bias by enforcing positive eigenvalues in GNN weight matrices to demonstrate the link between smoothness and robustness; (b) a Dirichlet energy overfitting control mechanism, which relies on a noise-free validation set; (c) a noise-robust loss function tailored for GNNs to induce smooth representations. Crucially, our methods do not degrade performance in noise-free data, reinforcing our central hypothesis that GNNs’ smoothness bias defines their robustness to label noise.

Primary Area: learning on graphs and other geometries & topologies

Submission Number: 20113