  Label-imbalanced and group-sensitive classification seeks to appropriately
modify standard training algorithms to optimize relevant metrics such as
balanced error and/or equal opportunity. For label imbalances, recent works
have proposed a logit-adjusted loss modification to standard empirical risk
minimization. We show that this might be ineffective in general and, in
particular so, in the overparameterized regime where training continues in the
zero training-error regime. Specifically for binary linear classification of a
separable dataset, we show that the modified loss converges to the max-margin
SVM classifier despite the logit adjustment. Instead, we propose a more general
vector-scaling loss that directly relates to the cost-sensitive SVM (CS-SVM),
thus favoring larger margin to the minority class. Through an insightful sharp
asymptotic analysis for a Gaussian-mixtures data model, we demonstrate the
efficacy of CS-SVM in balancing the errors of the minority/majority classes.
Our analysis also leads to a simple strategy for optimally tuning the involved
margin-ratio parameter. Then, we show how our results extend naturally to
binary classification with sensitive groups, thus treating the two common types
of imbalances (label/group) in a unifying way. We corroborate our theoretical
findings with numerical experiments on both synthetic and real-world datasets.
