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2021 Concentration inequalities for two-sample rank processes with application to bipartite ranking
Stephan Clémençon, Myrto Limnios, Nicolas Vayatis
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Electron. J. Statist. 15(2): 4659-4717 (2021). DOI: 10.1214/21-EJS1907


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.

Funding Statement

This work was supported with grants from Région Ile-de-France, the industrial chairs IDAML of Ecole Normale Supérieure Paris-Saclay and DSAIDIS of Télécom Paris, and ANR (project Limpid).


We thank the reviewers for their helpful comments.


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Stephan Clémençon. Myrto Limnios. Nicolas Vayatis. "Concentration inequalities for two-sample rank processes with application to bipartite ranking." Electron. J. Statist. 15 (2) 4659 - 4717, 2021.


Received: 1 April 2021; Published: 2021
First available in Project Euclid: 27 September 2021

Digital Object Identifier: 10.1214/21-EJS1907

Primary: 62G99 , 68Q32
Secondary: 60E15 , 62C12

Keywords: Bipartite ranking , Concentration inequalities , empirical risk minimization , generalization bounds , rank process , statistical learning theory , two-sample linear rank statistics


Vol.15 • No. 2 • 2021
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