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# S2: An Efficient Graph Based Active Learning Algorithm with Application to Nonparametric Classification

*Proceedings of The 28th Conference on Learning Theory*, PMLR 40:503-522, 2015.

#### Abstract

This paper investigates the problem of active learning for binary label prediction on a graph. We introduce a simple and label-efficient algorithm called S^2 for this task. At each step, S^2 selects the vertex to be labeled based on the structure of the graph and all previously gathered labels. Specifically, S^2 queries for the label of the vertex that bisects the \em shortest shortest path between any pair of oppositely labeled vertices. We present a theoretical estimate of the number of queries S^2 needs in terms of a novel parametrization of the complexity of binary functions on graphs. We also present experimental results demonstrating the performance of S^2 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 S^2 algorithm to the theory of nonparametric active learning. In particular, we show that S^2 achieves near minimax optimal excess risk for an important class of nonparametric classification problems.