Minimax Optimal Rates with Heavily Imbalanced Binary Data

In a wide range of binary prediction and estimation tasks, the data set exhibits a high degree of imbalance between the sample sizes of the two classes, which greatly hinders the performance of standard machine learning methods. In spite of a vast collection of methods aiming to achieve better performance on heavily imbalanced data, the theoretical limit of estimation with imbalanced data remains unknown. This paper provides some insights into the imbalanced classification problem by establishing the minimax risk of log-odds function estimation. Our minimax bounds reveal a notion of effective sample size. We further construct a sampling technique and prove that a minimax-rate optimal method for balanced data combined with the sampling technique achieves minimax-rate optimal performance on imbalanced data.