S²: An efficient graph based active learning algorithm with application to nonparametric classification

This paper investigates the problem of active learning for binary label prediction on a graph. We introduce a simple and label-efficient algorithm called S² for this task. At each step, S² selects the vertex to be labeled based on the structure of the graph and all previously gathered labels. Specifically, S² queries for the label of the vertex that bisects the shortest shortest path between any pair of oppositely labeled vertices. We present a theoretical estimate of the number of queries S² needs in terms of a novel parametrization of the complexity of binary functions on graphs. We also present experimental results demonstrating the performance of S² on both real and synthetic data. While other graph-based active learning algorithms have shown promise in practice, our algorithm is the first with both good performance and theoretical guarantees. Finally, we demonstrate the implications of the S² algorithm to the theory of nonparametric active learning. In particular, we show that S² achieves near minimax optimal excess risk for an important class of nonparametric classification problems.