Under-bagging (UB), which combines under sampling and bagging, is a popular ensemble learning method for training classifiers on an imbalanced data. Using bagging to reduce the increased variance caused by the reduction in sample size due to under sampling is a natural approach. However, it has recently been pointed out that in generalized linear models, naive bagging, which does not consider the class imbalance structure, and ridge regularization can produce the same results. Therefore, it is not obvious whether it is better to use UB, which requires an increased computational cost proportional to the number of under-sampled data sets, when training linear models. Given such a situation, in this study, we heuristically derive a sharp asymptotics of UB and use it to compare with several other standard methods for learning from imbalanced data, in the scenario where a linear classifier is trained from a two-component mixture data. The methods compared include the under-sampling (US) method, which trains a model using a single realization of the subsampled data, and the simple weighting (SW) method, which trains a model with a weighted loss on the entire data. It is shown that the performance of UB is improved by increasing the size of the majority class while keeping the size of the minority fixed, even though the class imbalance can be large, especially when the size of the minority class is small. This is in contrast to US, whose performance does not change as the size of the majority class increases, and SW, whose performance decreases as the imbalance increases. These results are different from the case of the naive bagging when training generalized linear models without considering the structure of the class imbalance, indicating the intrinsic difference between the ensembling and the direct regularization on the parameters.

翻译：Under-bagging（UB）结合了欠采样与Bagging技术，是一种用于在不平衡数据集上训练分类器的常用集成学习方法。通过Bagging来降低因欠采样导致样本量减少所引发的方差增加，是一种自然的策略。然而，近期有研究指出，在广义线性模型中，未考虑类别不平衡结构的朴素Bagging与岭正则化可产生相同的结果。因此，在训练线性模型时，使用需要耗费与欠采样数据集数量成正比的额外计算成本的UB是否更优，并不明确。基于此背景，本研究启发式地推导了UB的锐利渐近性质，并将其与若干处理不平衡数据学习的标准方法进行对比，场景设定为从双组分混合数据中训练线性分类器。对比方法包括：欠采样（US）方法（基于单次子采样数据训练模型）和简单加权（SW）方法（在全体数据上使用加权损失训练模型）。结果表明，通过固定少数类样本量、增加多数类样本量，UB的性能可得到提升，即便在不平衡程度较大时也是如此，尤其在少数类样本量较小的情形下。这与US（其性能不随多数类样本量增加而改变）及SW（其性能随不平衡程度加剧而下降）的情况形成鲜明对比。这些结果与未考虑类别不平衡结构时在广义线性模型上训练的朴素Bagging情况不同，揭示了集成方法与直接对参数进行正则化之间的本质差异。