Concentration Inequalities for Two-Sample Rank Processes with Application to Bipartite Ranking

TL;DR

This paper develops concentration inequalities for two-sample rank processes, providing theoretical bounds on their behavior, and applies these results to analyze the generalization performance of ranking metrics like AUC, supported by numerical experiments.

Contribution

It introduces new concentration inequalities for two-sample rank processes and applies them to assess the generalization of ranking performance measures.

Findings

Derived nonasymptotic concentration bounds for two-sample rank processes.

Established theoretical guarantees for empirical maximizers of ranking criteria.

Validated results with numerical experiments demonstrating practical relevance.

Abstract

The ROC curve is the gold standard for measuring the performance of a test/scoring statistic regarding its capacity to discriminate between two statistical populations in a wide variety of applications, ranging from anomaly detection in signal processing to information retrieval, through medical diagnosis. Most practical performance measures used in scoring/ranking applications such as the AUC, the local AUC, the p-norm push, the DCG and others, can be viewed as summaries of the ROC curve. In this paper, the fact that most of these empirical criteria can be expressed as two-sample linear rank statistics is highlighted and concentration inequalities for collections of such random variables, referred to as two-sample rank processes here, are proved, when indexed by VC classes of scoring functions. Based on these nonasymptotic bounds, the generalization capacity of empirical maximizers of a wide class of ranking performance criteria is next investigated from a theoretical perspective. It is also supported by empirical evidence through convincing numerical experiments.