How Does Knowledge of the AUC Constrain the Set of Possible Ground-truth Labelings?

TL;DR

This paper investigates how knowing the AUC score in binary classification can limit the possible true labelings, providing algorithms to count and enumerate these labelings and revealing that the number of compatible labelings can decrease with larger test sets.

Contribution

It introduces a mathematical framework for understanding how AUC constrains ground-truth labels and presents algorithms for counting and enumerating these labelings.

Findings

Knowledge of AUC constrains possible labelings.

Algorithms for exact counting and enumeration of labelings.

Number of compatible labelings can decrease as test set size increases.

Abstract

Recent work on privacy-preserving machine learning has considered how data-mining competitions such as Kaggle could potentially be "hacked", either intentionally or inadvertently, by using information from an oracle that reports a classifier's accuracy on the test set. For binary classification tasks in particular, one of the most common accuracy metrics is the Area Under the ROC Curve (AUC), and in this paper we explore the mathematical structure of how the AUC is computed from an n-vector of real-valued "guesses" with respect to the ground-truth labels. We show how knowledge of a classifier's AUC on the test set can constrain the set of possible ground-truth labelings, and we derive an algorithm both to compute the exact number of such labelings and to enumerate efficiently over them. Finally, we provide empirical evidence that, surprisingly, the number of compatible labelings can actually decrease as n grows, until a test set-dependent threshold is reached.