We generate a training data set with $5$ minority samples and $200$ majority samples. The estimated decision sets of unpruned CART and optimally pruned CART [@breiman1984classification] are shown in Figure (a) and (b), respectively. Minority samples are given weight 32 while majority samples are given weight 1 to naively address the imbalance. Both decision sets are inaccurate, with unpruned CART overfitting and pruned CART also poor.

Imbalanced data problems have drawn substantial interest; see [@haixiang2017learning], [@he2008learning], [@krawczyk2016learning] and [@fernandez2017insight] for reviews. Some early work relied on random under- or over-sampling, which is essentially equivalent to modifying the weights and cannot address the key problem. [@chawla2002smote] proposed SMOTE, which instead creates synthetic samples. For each minority class sample, they create synthetic samples along the line segments that join each minority class sample with its k nearest neighbors in the minority class. Building on this idea, many other synthetic sampling methods have been proposed, including ADASYN [@he2008adasyn], Borderline-SMOTE [@han2005borderline], SPIDER [@stefanowski2008selective], safe-level-SMOTE [@bunkhumpornpat2009safe] and WGAN-Based sampling [@wang2019wgan]. These synthetic sampling methods have been demonstrated to be relatively effective.

However, current understanding of synthetic sampling is inadequate. [@chawla2002smote] motivates SMOTE as designed to “create large and less specific decision regions”, “rather than smaller 