  In most machine learning applications, classification accuracy is not the primary metric of interest. Binary classifiers which face class imbalance are often evaluated by the $F_\beta$ score, area under the precision-recall curve, Precision at K, and more. The maximization of many of these metrics can be expressed as a constrained optimization problem, where the constraint is a function of the classifier's predictions.   In this paper we propose a novel framework for learning with constraints that can be expressed as a predicted positive rate (or negative rate) on a subset of the training data. We explicitly model the threshold at which a classifier must operate to satisfy the constraint, yielding a surrogate loss function which avoids the complexity of constrained optimization. The method is model-agnostic and only marginally more expensive than minimization of the unconstrained loss. Experiments on a variety of benchmarks show competitive performance relative to existing baselines. 
Constrained Classification and Ranking via Quantiles