On the Expressiveness and Learning of Relational Neural Networks on Hypergraphs
This paper presents a framework for analyzing the expressiveness and learning of relational models applied to hypergraph reasoning tasks. We start with a general framework that unifies several relational neural network architectures: graph neural networks, neural logical machines, and transformers. Our first contribution is a fine-grained analysis of the expressiveness of these neural networks, that is, the set of functions that they can realize and the set of problems that they can solve. Our result is a hierarchy of problems they can solve, defined in terms of various hyperparameters such as depth and width. Next, we analyze the learning properties of these neural networks, especially focusing on how they can be trained on a small graphs and generalize to larger graphs. Our theoretical results are further supported by the empirical results illustrating the optimization and generalization of these models based on gradient-descent training.
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