# Generative Risk Minimization for Out-of-Distribution Generalization on Graphs

arXiv: https://arxiv.org/abs/2502.07968 · HTML: https://mlsanity.com/paper/2502.07968/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 58 of its 72 references that are in the map.
- First posted on arXiv 11 Feb 2025.

## Builds on (most cited first, 30 of 58)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Representation Learning with Contrastive Predictive Coding](https://mlsanity.com/paper/1807.03748/) (2018)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [mixup: Beyond Empirical Risk Minimization](https://mlsanity.com/paper/1710.09412/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Supervised Contrastive Learning](https://mlsanity.com/paper/2004.11362/) (2020)
- [Open Graph Benchmark: Datasets for Machine Learning on Graphs](https://mlsanity.com/paper/2005.00687/) (2020)
- [Unsupervised Domain Adaptation by Backpropagation](https://mlsanity.com/paper/1409.7495/) (2014)
- [Invariant Risk Minimization](https://mlsanity.com/paper/1907.02893/) (2019)
- [Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/1812.08434/) (2018)
- [Do ImageNet Classifiers Generalize to ImageNet?](https://mlsanity.com/paper/1902.10811/) (2019)
- [Variational Graph Auto-Encoders](https://mlsanity.com/paper/1611.07308/) (2016)
- [WILDS: A Benchmark of in-the-Wild Distribution Shifts](https://mlsanity.com/paper/2012.07421/) (2020)
- [Moment Matching for Multi-Source Domain Adaptation](https://mlsanity.com/paper/1812.01754/) (2018)
- [MoleculeNet: A Benchmark for Molecular Machine Learning](https://mlsanity.com/paper/1703.00564/) (2017)
- [An Introduction to Variational Autoencoders](https://mlsanity.com/paper/1906.02691/) (2019)
- [Out-of-Distribution Generalization via Risk Extrapolation (REx)](https://mlsanity.com/paper/2003.00688/) (2020)
- [GNNExplainer: Generating Explanations for Graph Neural Networks](https://mlsanity.com/paper/1903.03894/) (2019)
- [Recognition in Terra Incognita](https://mlsanity.com/paper/1807.04975/) (2018)
- [Junction Tree Variational Autoencoder for Molecular Graph Generation](https://mlsanity.com/paper/1802.04364/) (2018)
- [Domain Generalization via Invariant Feature Representation](https://mlsanity.com/paper/1301.2115/) (2013)
- [One pixel attack for fooling deep neural networks](https://mlsanity.com/paper/1710.08864/) (2017)
- [GraphRNN: Generating Realistic Graphs with Deep Auto-regressive Models](https://mlsanity.com/paper/1802.08773/) (2018)
- [MolGAN: An implicit generative model for small molecular graphs](https://mlsanity.com/paper/1805.11973/) (2018)
- [Return of Frustratingly Easy Domain Adaptation](https://mlsanity.com/paper/1511.05547/) (2015)
- [Explainability in Graph Neural Networks: A Taxonomic Survey](https://mlsanity.com/paper/2012.15445/) (2020)
- [GraphVAE: Towards Generation of Small Graphs Using Variational Autoencoders](https://mlsanity.com/paper/1802.03480/) (2018)
- [Environment Inference for Invariant Learning](https://mlsanity.com/paper/2010.07249/) (2020)
- [Fr'echet ChemNet Distance: A metric for generative models for molecules in drug discovery](https://mlsanity.com/paper/1803.09518/) (2018)
- [Invariant Risk Minimization Games](https://mlsanity.com/paper/2002.04692/) (2020)

## Built on by (most cited first, 2 of 2)

- [EdgeMask-DG*: Learning Domain-Invariant Graph Structures via Adversarial Edge Masking](https://mlsanity.com/paper/2602.05571/) (2026)
- [On the Safety of Graph Representation Learning](https://mlsanity.com/paper/2605.06576/) (2026)

## Abstract (arXiv)

Out-of-distribution (OOD) generalization on graphs aims at dealing with scenarios where the test graph distribution differs from the training graph distributions. Compared to i.i.d. data like images, the OOD generalization problem on graph-structured data remains challenging due to the non-i.i.d. property and complex structural information on graphs. Recently, several works on graph OOD generalization have explored extracting invariant subgraphs that share crucial classification information across different distributions. Nevertheless, such a strategy could be suboptimal for entirely capturing the invariant information, as the extraction of discrete structures could potentially lead to the loss of invariant information or the involvement of spurious information. In this paper, we propose an innovative framework, named Generative Risk Minimization (GRM), designed to generate an invariant subgraph for each input graph to be classified, instead of extraction. To address the challenge of optimization in the absence of optimal invariant subgraphs (i.e., ground truths), we derive a tractable form of the proposed GRM objective by introducing a latent causal variable, and its effectiveness is validated by our theoretical analysis. We further conduct extensive experiments across a variety of real-world graph datasets for both node-level and graph-level OOD generalization, and the results demonstrate the superiority of our framework GRM.