Uncertainty Modeling in Graph Neural Networks via Stochastic Differential Equations (2408.16115v4)

Abstract: We propose a novel Stochastic Differential Equation (SDE) framework to address the problem of learning uncertainty-aware representations for graph-structured data. While Graph Neural Ordinary Differential Equations (GNODEs) have shown promise in learning node representations, they lack the ability to quantify uncertainty. To address this, we introduce Latent Graph Neural Stochastic Differential Equations (LGNSDE), which enhance GNODE by embedding randomness through a Bayesian prior-posterior mechanism for epistemic uncertainty and Brownian motion for aleatoric uncertainty. By leveraging the existence and uniqueness of solutions to graph-based SDEs, we prove that the variance of the latent space bounds the variance of model outputs, thereby providing theoretically sensible guarantees for the uncertainty estimates. Furthermore, we show mathematically that LGNSDEs are robust to small perturbations in the input, maintaining stability over time. Empirical results across several benchmarks demonstrate that our framework is competitive in out-of-distribution detection, robustness to noise, and active learning, underscoring the ability of LGNSDEs to quantify uncertainty reliably.

• The paper introduces LGNSDE to quantify both aleatoric and epistemic uncertainties in graph neural networks.
• It leverages Brownian motion through stochastic differential equations to robustly capture data noise.
• Empirical results show LGNSDE outperforms Bayesian GCNs and ensembles, improving OOD detection and overall reliability.

Uncertainty Modeling in Graph Neural Networks via Stochastic Differential Equations

The paper "Uncertainty Modeling in Graph Neural Networks via Stochastic Differential Equations" addresses a notable gap in the field of Graph Neural Networks (GNNs) related to the quantification of uncertainty in node representations. Despite GNNs' efficacy in various applications, from social network analysis to molecular biology, their inability to effectively quantify uncertainty poses significant limitations. This paper introduces Latent Graph Neural Stochastic Differential Equations (LGNSDE) as a promising solution for embedding randomness and quantifying both aleatoric and epistemic uncertainties in graph-structured data.

Background and Motivation

Traditional differential equations have long been pivotal in modeling complex systems across multiple scientific fields. However, with the rise of neural networks, GNNs have become the preferred tool for handling graph-structured data due to their capability to capture intricate relationships between nodes. Nevertheless, GNNs lack the ability to reliably quantify uncertainty, which is critical for decision-making processes in applications requiring high reliability. To address this, the authors propose a novel approach integrating Stochastic Differential Equations (SDEs) with GNNs, enhancing the framework with a probabilistic component to model uncertainty more effectively than existing methods like Bayesian GCNs and GCN ensembles.

Methodology

The crux of the proposed method, LGNSDE, lies in leveraging SDEs to incorporate randomness into node features through Brownian motion during both training and inference. This approach enables the model to capture the inherent noise (aleatoric uncertainty) in the data. Moreover, by employing a prior SDE latent space and learning a posterior SDE representation via a GNN, LGNSDE can quantify the model uncertainty (epistemic uncertainty) resulting from limited data.

Key contributions of the methodology include:

1. Novel Model Design: LGNSDE combines the robustness of SDEs with the flexibility of GNNs for handling graph-structured data, effectively capturing both epistemic and aleatoric uncertainties.
2. Theoretical Guarantees: The authors provide theoretical underpinnings demonstrating meaningful uncertainty estimates and robustness to input perturbations.
3. Empirical Validation: The effectiveness of LGNSDE in uncertainty quantification is empirically validated, showing superior performance compared to Bayesian GCNs and GCN ensembles.

Theoretical Framework

The paper presents rigorous theoretical guarantees to support the stability and robustness of LGNSDE. Under certain assumptions, the authors establish that the variance of the latent representation bounds the variance of the model output, providing a meaningful measure of the total uncertainty. Further, they derive bounds on the deviation between solutions with perturbed initial conditions, ensuring the model's robustness over time.

Experimental Results

Evaluations were conducted on five datasets, including well-known benchmarks such as CORA and Citeseer. LGNSDE consistently outperforms or matches the performance of GNODE, GCN, Bayesian GCN, and GCN ensemble models across key metrics, including Micro-AUROC, AURC, and accuracy. Notably, LGNSDE demonstrates:

• Higher AUROC and Accuracy: Indicating robust classification capabilities and power in distinguishing between classes.
• Lower AURC: Reflecting a strong ability to maintain low risk while ensuring confident predictions.

Additionally, the LGNSDE model exhibits significant improvements in out-of-distribution (OOD) detection, an essential metric for assessing a model's reliability in real-world scenarios. The LGNSDE model shows a clear separation between in-distribution and OOD data, with considerably higher entropy for OOD samples, showcasing its superior uncertainty quantification.

Practical and Theoretical Implications

The implications of LGNSDE are profound both practically and theoretically. Practically, it enhances the reliability of GNNs in critical applications such as risk assessment and resource allocation where uncertainty quantification is paramount. Theoretically, it advances the integration of probabilistic methods within deep learning frameworks, specifically in graph-based domains.

Future Directions

The challenges of time and memory complexity in neural SDEs open avenues for future research. Optimizing sampling methods to make the approach more scalable remains a critical direction. Furthermore, expanding benchmarks to provide a more comprehensive evaluation will solidify the practical applications of LGNSDE.

In conclusion, this paper makes a significant contribution by bridging the gap between stochastic processes and graph neural networks, offering a robust framework for uncertainty modeling in graph-structured data. The approach is backed by solid theoretical foundations and compelling empirical results.