  Mitigating the disparate impact of statistical machine learning methods is
crucial for ensuring fairness. While extensive research aims to reduce
disparity, the effect of using a \emph{finite dataset} -- as opposed to the
entire population -- remains unclear. This paper explores the statistical
foundations of fair binary classification with two protected groups, focusing
on controlling demographic disparity, defined as the difference in acceptance
rates between the groups. Although fairness may come at the cost of accuracy
even with infinite data, we show that using a finite sample incurs additional
costs due to the need to estimate group-specific acceptance thresholds. We
study the minimax optimal classification error while constraining demographic
disparity to a user-specified threshold. To quantify the impact of fairness
constraints, we introduce a novel measure called \emph{fairness-aware excess
risk} and derive a minimax lower bound on this measure that all classifiers
must satisfy. Furthermore, we propose FairBayes-DDP+, a group-wise thresholding
method with an offset that we show attains the minimax lower bound. Our lower
bound proofs involve several innovations. Experiments support that
FairBayes-DDP+ controls disparity at the user-specified level, while being
faster and having a more favorable fairness-accuracy tradeoff than several
baselines.
