Ranking and Empirical Minimization of U-Statistics

The problem of ranking/ordering instances, instead of simply classifying them, has recently gained much attention in machine learning. In this paper we formulate the ranking problem in a rigorous statistical framework. The goal is to learn a ranking rule for deciding, among two instances, which one...

Ausführliche Beschreibung

Bibliographische Detailangaben
Veröffentlicht in:	The Annals of Statistics. - Institute of Mathematical Statistics. - 36(2008), 2, Seite 844-874
1. Verfasser:	Clémençon, Stéphan (VerfasserIn)
Weitere Verfasser:	Lugosi, Gábor, Vayatis, Nicolas
Format:	Online-Aufsatz
Sprache:	English
Veröffentlicht:	2008
Zugriff auf das übergeordnete Werk:	The Annals of Statistics
Schlagworte:	Statistical learning Theory of classification VC classes Fast rates Convex risk minimization Moment inequalities U-processes Mathematics Physical sciences Applied sciences mehr... Behavioral sciences Economics


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520			\|a The problem of ranking/ordering instances, instead of simply classifying them, has recently gained much attention in machine learning. In this paper we formulate the ranking problem in a rigorous statistical framework. The goal is to learn a ranking rule for deciding, among two instances, which one is "better," with minimum ranking risk. Since the natural estimates of the risk are of the form of a U-statistic, results of the theory of U-processes are required for investigating the consistency of empirical risk minimizers. We establish, in particular, a tail inequality for degenerate U-processes, and apply it for showing that fast rates of convergence may be achieved under specific noise assumptions, just like in classification. Convex risk minimization methods are also studied.
540			\|a Copyright 2008 The Institute of Mathematical Statistics
650		4	\|a Statistical learning
650		4	\|a Theory of classification
650		4	\|a VC classes
650		4	\|a Fast rates
650		4	\|a Convex risk minimization
650		4	\|a Moment inequalities
650		4	\|a U-processes
650		4	\|a Mathematics \|x Pure mathematics \|x Probability theory \|x Random variables
650		4	\|a Physical sciences \|x Physics \|x Mechanics \|x Wave mechanics \|x Waves \|x Wave phenomena \|x Wave scattering \|x Electromagnetic scattering \|x Umklapp process
650		4	\|a Applied sciences \|x Computer science \|x Artificial intelligence \|x Machine learning
650		4	\|a Mathematics \|x Applied mathematics \|x Statistics
650		4	\|a Mathematics \|x Mathematical procedures \|x Approximation
650		4	\|a Behavioral sciences \|x Psychology \|x Cognitive psychology \|x Cognitive processes \|x Learning \|x Learning theory
650		4	\|a Economics \|x Microeconomics \|x Economic costs and benefits \|x Economic costs \|x Cost functions
650		4	\|a Mathematics \|x Applied mathematics \|x Statistics \|x Applied statistics \|x Descriptive statistics \|x Measures of variability \|x Statistical variance
650		4	\|a Mathematics \|x Mathematical values \|x Mathematical constants
650		4	\|a Mathematics \|x Pure mathematics \|x Linear algebra \|x Vector analysis \|x Mathematical vectors
650		4	\|a Mathematics \|x Pure mathematics \|x Probability theory \|x Random variables
650		4	\|a Physical sciences \|x Physics \|x Mechanics \|x Wave mechanics \|x Waves \|x Wave phenomena \|x Wave scattering \|x Electromagnetic scattering \|x Umklapp process
650		4	\|a Applied sciences \|x Computer science \|x Artificial intelligence \|x Machine learning
650		4	\|a Mathematics \|x Applied mathematics \|x Statistics
650		4	\|a Mathematics \|x Mathematical procedures \|x Approximation
650		4	\|a Behavioral sciences \|x Psychology \|x Cognitive psychology \|x Cognitive processes \|x Learning \|x Learning theory
650		4	\|a Economics \|x Microeconomics \|x Economic costs and benefits \|x Economic costs \|x Cost functions
650		4	\|a Mathematics \|x Applied mathematics \|x Statistics \|x Applied statistics \|x Descriptive statistics \|x Measures of variability \|x Statistical variance
650		4	\|a Mathematics \|x Mathematical values \|x Mathematical constants
650		4	\|a Mathematics \|x Pure mathematics \|x Linear algebra \|x Vector analysis \|x Mathematical vectors
655		4	\|a research-article
700	1		\|a Lugosi, Gábor \|e verfasserin \|4 aut
700	1		\|a Vayatis, Nicolas \|e verfasserin \|4 aut
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