Learning with Small Data: Subgraph Counting Queries

Deep Learning (DL) has been widely used in many applications, and its success is achieved with large training data. A key issue is how to provide a DL solution when there is no large training data to learn initially. In this paper, we explore a meta-learning approach for a specific problem, subgraph isomorphism counting, which is a fundamental problem in graph analysis to count the number of a given pattern graph, p , in a data graph, g , that matches p . There are various data graphs and pattern graphs. A subgraph isomorphism counting query is specified by a pair, ( g , p ). This problem is NP-hard and needs large training data to learn by DL in nature. We design a Gaussian Process (GP) model which combines Graph Neural Network with Bayesian nonparametric, and we train the GP by a meta-learning algorithm on a small set of training data. By meta-learning, we can obtain a generalized meta-model to better encode the information of data and pattern graphs and capture the prior of small tasks. With the meta-model learned, we handle a collection of pairs ( g , p ), as a task, where some pairs may be associated with the ground-truth, and some pairs are the queries to answer. There are two cases. One is there are some with ground-truth (few-shot), and one is there is none with ground-truth (zero-shot). We provide our solutions for both. In particular, for zero-shot, we propose a new data-driven approach to predict the count values. Note that zero-shot learning for our regression tasks is difficult, and there is no hands-on solution in the literature. We conducted extensive experimental studies to confirm that our approach is robust to model degeneration on small training data, and our meta-model can fast adapt to new queries by few-shot and zero-shot learning.