Data-Driven Estimation of the False Positive Rate of the Bayes Binary Classifier via Soft Labels

Classification is a fundamental task in many applications on which data-driven methods have shown outstanding performances. However, it is challenging to determine whether such methods have achieved the optimal performance. This is mainly because the best achievable performance is typically unknown and hence, effectively estimating it is of prime importance. In this paper, we consider binary classification problems and we propose an estimator for the false positive rate (FPR) of the Bayes classifier, that is, the optimal classifier with respect to accuracy, from a given dataset. Our method utilizes soft labels, or real-valued labels, which are gaining significant traction thanks to their properties. We thoroughly examine various theoretical properties of our estimator, including its consistency, unbiasedness, rate of convergence, and variance. To enhance the versatility of our estimator beyond soft labels, we also consider noisy labels, which encompass binary labels. For noisy labels, we develop effective FPR estimators by leveraging a denoising technique and the Nadaraya-Watson estimator. Due to the symmetry of the problem, our results can be readily applied to estimate the false negative rate of the Bayes classifier.