The Annals of Statistics

Local Rademacher complexities and oracle inequalities in risk minimization

Vladimir Koltchinskii

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Let ℱ be a class of measurable functions f:S↦[0, 1] defined on a probability space (S, $\mathcal{A}$, P). Given a sample (X1, …, Xn) of i.i.d. random variables taking values in S with common distribution P, let Pn denote the empirical measure based on (X1, …, Xn). We study an empirical risk minimization problem Pnf→min , f∈ℱ. Given a solution n of this problem, the goal is to obtain very general upper bounds on its excess risk

\[\mathcal{E}_{P}(\hat{f}_{n}):=P\hat{f}_{n}-\inf_{f\in \mathcal{F}}Pf,\]

expressed in terms of relevant geometric parameters of the class ℱ. Using concentration inequalities and other empirical processes tools, we obtain both distribution-dependent and data-dependent upper bounds on the excess risk that are of asymptotically correct order in many examples. The bounds involve localized sup-norms of empirical and Rademacher processes indexed by functions from the class. We use these bounds to develop model selection techniques in abstract risk minimization problems that can be applied to more specialized frameworks of regression and classification.

Article information

Ann. Statist., Volume 34, Number 6 (2006), 2593-2656.

First available in Project Euclid: 23 May 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20] 60B99: None of the above, but in this section 68Q32: Computational learning theory [See also 68T05]
Secondary: 62G08: Nonparametric regression 68T05: Learning and adaptive systems [See also 68Q32, 91E40] 68T10: Pattern recognition, speech recognition {For cluster analysis, see 62H30}

Rademacher complexities empirical risk minimization oracle inequalities model selection concentration inequalities classification


Koltchinskii, Vladimir. Local Rademacher complexities and oracle inequalities in risk minimization. Ann. Statist. 34 (2006), no. 6, 2593--2656. doi:10.1214/009053606000001019.

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