We study the problem of fair binary classification using the notion of Equal Opportunity. It requires the true positive rate to distribute equally across the sensitive groups. Within this setting we show that the fair optimal classifier is obtained by recalibrating the Bayes classifier by a group-dependent threshold. We provide a constructive expression for the threshold. This result motivates us to devise a plug-in classification procedure based on both unlabeled and labeled datasets. While the latter is used to learn the output conditional probability, the former is used for calibration. The overall procedure can be computed in polynomial time and it is shown to be statistically consistent both in terms of classification error and fairness measure. Finally, we present numerical experiments which indicate that our method is often superior or competitive with the state-of-the-art methods on benchmark datasets.
A class of nonparametric regression function estimates generalizing the nearest neighbor estimate of Cover [ 12] is presented. Under various noise conditions, it is shown that the estimates are strongly uniformly consistent. The uniform convergence of the estimates can be exploited to design a simple random search algorithm for the global minimization of the regression function. It has been recently shown that, under the margin (or low noise) assumption, there exist classifiers attaining fast rates of convergence of the excess Bayes risk, that is, rates faster than n -1 2 . The work on this subject has suggested the following two conjectures: (i) the best achievable fast rate is of the order n -1 , and (ii) the plug-in classifiers generally converge more slowly than the classifiers based on empirical risk minimization. We show that both conjectures are not correct. In particular, we construct plug-in classifiers that can achieve not only fast, but also super-fast rates, that is, rates faster than n -1 . We establish minimax lower bounds showing that the obtained rates cannot be improved. 