  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 popular 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 under-sampled 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 is almost independent of the majority class size. In this sense,
bagging and simple regularization differ as methods to reduce the variance
increased by under-sampling. On the other hand, the performance of SW with the
optimal weighting coefficients is almost equal to UB, indicating that the
combination of reweighting and regularization may be similar to UB.
