Distribution Matching for Graph Quantification Under Structural Covariate Shift

Graphs are commonly used in machine learning to model relationships between instances. Consider the task of predicting the political preferences of users in a social network; to solve this task one should consider, both, the features of each individual user and the relationships between them. However, oftentimes one is not interested in the label of a single instance but rather in the distribution of labels over a set of instances; e.g., when predicting the political preferences of users, the overall prevalence of a given opinion might be of higher interest than the opinion of a specific person. This label prevalence estimation task is commonly referred to as quantification learning (QL). Current QL methods for tabular data are typically based on the so-called prior probability shift (PPS) assumption which states that the label-conditional instance distributions should remain equal across the training and test data. In the graph setting, PPS generally does not hold if the shift between training and test data is structural, i.e., if the training data comes from a different region of the graph than the test data. To address such structural shifts, an importance sampling variant of the popular adjusted count quantification approach has previously been proposed. In this work, we extend the idea of structural importance sampling to the state-of-the-art KDEy quantification approach. We show that our proposed method adapts to structural shifts and outperforms standard quantification approaches.