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Quantile regression, based on check loss, is a widely used inferentialparadigm in Econometrics and Statistics. The conditional quantiles provide arobust alternative to classical conditional means, and also allow uncertaintyquantification of the predictions, while making very few distributionalassumptions. We consider the analogue of check loss in the binaryclassification setting. We assume that the conditional quantiles are smoothfunctions that can be learnt by Deep Neural Networks (DNNs). Subsequently, wecompute the Lipschitz constant of the proposed loss, and also show that itscurvature is bounded, under some regularity conditions. Consequently, recentresults on the error rates and DNN architecture complexity become directlyapplicable.  We quantify the uncertainty of the class probabilities in terms of predictionintervals, and develop individualized confidence scores that can be used todecide whether a prediction is reliable or not at scoring time. By aggregatingthe confidence scores at the dataset level, we provide two additional metrics,model confidence, and retention rate, to complement the widely used classifiersummaries. We also the robustness of the proposed non-parametric binaryquantile classification framework are also studied, and we demonstrate how toobtain several univariate summary statistics of the conditional distributions,in particular conditional means, using smoothed conditional quantiles, allowingthe use of explanation techniques like Shapley to explain the mean predictions.Finally, we demonstrate an efficient training regime for this loss based onStochastic Gradient Descent with Lipschitz Adaptive Learning Rates (LALR).

Estimation and Applications of Quantiles in Deep Binary Classification