Shows Graph Attention Networks Overcome Noise in Random Geometric Graphs with Erdős–Rényi Contamination

Researchers are increasingly reliant on graph-structured data, yet establishing the statistical benefits of graph attention networks (GATs) over conventional machine learning methods remains a challenge. Somak Laha from Harvard University, Suqi Liu from the University of California, Riverside, and Morgane Austern, also of Harvard University, alongside their colleagues, tackle this problem by investigating node regression on random geometric graphs corrupted by Erdős, Rényi noise. Their work is significant because it provides rigorous theoretical guarantees demonstrating that a carefully designed, task-specific GAT outperforms both ordinary least squares and standard graph convolutional networks in estimating regression coefficients and predicting responses, even with noisy data and imperfect graph structures. This analysis, supported by synthetic and real-world experiments, offers valuable insight into the robustness and advantages of GATs for statistical inference on graphs.

Statistical benefits of graph attention networks with noisy covariates and edges are now becoming clear

Scientists have demonstrated a provable advantage of graph attention networks (GATs) over non-attention graph neural networks (GNNs) for node regression, addressing a long-standing gap in rigorous statistical guarantees. The research focuses on a challenging scenario involving simultaneous covariate and edge corruption, where responses are generated from latent node-level covariates, but only noise-perturbed versions are observed.
The team proposes and analyses a task-specific GAT designed to construct denoised proxy features for regression, operating on a random geometric graph contaminated by independent Erdős, Rényi edges. This carefully designed GAT achieves lower error asymptotically in estimating the regression coefficient compared to ordinary least squares (OLS) on noisy node covariates, and in predicting the response for an unlabelled node compared to a vanilla graph convolutional network (GCN), under mild growth conditions.

The study establishes a novel framework for analysing GATs by leveraging high-dimensional geometric tail bounds and concentration for neighbourhood counts and sample covariances. Researchers designed a discrete-attention scheme to yield consistent estimation and lower prediction error than non-attention message passing, specifically in a node-regression task with noisy graph side information.

The work models a network of nodes where responses are influenced by unobserved latent covariates, with observed features being noise-amplified versions of these latent variables. The underlying graph structure is a random geometric graph, further corrupted by independent Erdős, Rényi edges, creating a complex interplay between signal and noise.

Experiments on synthetically generated data verify the theoretical findings, demonstrating the effectiveness of the proposed GAT in mitigating the attenuation bias inherent in errors-in-variables problems. Further validation is provided through experiments on real-world graphs, showcasing the attention mechanism’s effectiveness in several node regression tasks.

The research proves that regressing on denoised proxies, constructed via a two-layer attention-based GNN, yields consistent estimation for the regression coefficient. Moreover, the proposed method achieves strictly smaller asymptotic risk compared to finite-depth GNNs when Erdős, Rényi edges dominate geometric ones, opening avenues for more robust and accurate graph-based learning.

Denoising node covariates and graph structure via a discretised attention mechanism improves graph representation learning

Scientists developed a novel graph attention network (GAT) to address challenges in node regression under conditions of both covariate and edge corruption. The study focused on scenarios where observed node covariates are noise-perturbed versions of latent variables and the graph structure itself is contaminated by spurious edges.

Researchers constructed a task-specific GAT designed to create denoised proxy features for improved regression accuracy. To overcome the attenuation bias inherent in ordinary least squares (OLS) regression with noisy covariates, the team engineered a discretized attention mechanism. This mechanism computes node-level denoised proxies, λi, representing estimates of the latent variables, xi.

Experiments employed a two-layer attention-based graph neural network to selectively average neighbour covariates, constructing these proxies. The innovative approach leverages the principle that geometrically close neighbours tend to share similar latent covariates, enabling denoising through message passing along geometric edges.

Crucially, the study pioneered a cross-fitting attention rule to decouple feature selection from the averaged coordinates. Each observed covariate, zi, was split into two disjoint coordinate blocks; dot-products within one block determined neighbour selection, while averaging occurred using the other block.

This design minimized bias by preventing correlation between the selection event and the regression residuals. The resulting proxies, collected in matrix Λn, were then used in a final OLS regression to estimate the regression coefficient, β. Analysis demonstrates that, under mild growth conditions relating the number of nodes, dimensionality, and graph characteristics, the attention-based proxies closely approximate the latent covariates.

Consequently, the OLS estimate of β becomes consistent and achieves a lower asymptotic mean squared error (MSE) compared to finite-depth graph neural networks that aggregate raw neighbourhoods, particularly when spurious edges dominate. Synthetic data experiments verified these theoretical findings, while real-world graph analyses demonstrated the effectiveness of the method across several node regression tasks.

Theoretical robustness and improved coefficient estimation with graph attention networks are key benefits

Scientists have demonstrated a provable advantage of a carefully designed Graph Attention Network (GAT) over non-graph neural networks for node regression under conditions of both covariate and edge corruption. The research addresses a gap in rigorous statistical guarantees for GAT robustness, focusing on scenarios where observed node covariates are noise-perturbed versions of latent values and the graph structure is contaminated by independent Erdős, Rényi edges.

Experiments utilising synthetically generated data verified these theoretical findings. The team proved that regressing response variables on denoised proxy features, constructed by the task-specific GAT, achieves lower error asymptotically in estimating the regression coefficient compared to ordinary least squares (OLS) on noisy node covariates.

Measurements confirm this improvement in coefficient estimation, demonstrating the GAT’s ability to overcome the attenuation bias faced by OLS. Furthermore, the GAT demonstrably outperforms a vanilla Graph Convolutional Network (GCN) in predicting the response for an unlabelled node. Results demonstrate that the GAT-style model provably outperforms all non-attention graph neural networks under the same data generative assumptions.

The analysis leverages high-dimensional geometric tail bounds and concentration for neighbourhood counts and sample covariances, enabling precise quantification of performance gains. Specifically, the study shows that the Erdős, Rényi degree dominates the geometric degree in the observed graph. Tests prove the effectiveness of the approach in several node regression tasks on both synthetic and real-world graphs.

The work establishes that the GAT achieves consistent estimation of coefficients for node regression, even with constant feature-noise levels and under Erdős, Rényi structural contamination. This breakthrough delivers a robust solution for node regression in noisy environments, offering potential for applications in areas where data corruption is prevalent.

Statistical benefits of graph attention networks under covariate and edge noise remain largely unexplored

Researchers have demonstrated that a carefully designed graph attention network (GAT) can outperform standard graph convolutional networks (GCNs) in node regression tasks, even when faced with noisy data. This work addresses a gap in understanding the statistical advantages of GATs by analysing performance under conditions of both covariate and edge corruption, modelling responses generated from latent node covariates with observed, noise-perturbed versions.

The analysis centres on a task-specific GAT that constructs denoised proxy features for regression, achieving lower error in estimating regression coefficients and predicting responses for unlabelled nodes compared to ordinary least squares and vanilla GCNs respectively. The significance of these findings lies in establishing provable benefits of GATs under realistic conditions of data imperfection.

By leveraging high-dimensional geometric tail bounds and concentration inequalities, the researchers provide theoretical guarantees for the GAT’s superior performance, verified through experiments on both synthetic and real-world graphs. These results suggest that GATs are not merely empirically successful but possess inherent statistical properties that make them more robust to noise and more accurate in node regression.

The authors acknowledge limitations stemming from the mild growth conditions required for their theoretical results and the specific error models considered. Future research could explore the extension of these findings to more complex graph structures and noise distributions, potentially broadening the applicability of this task-specific GAT design.