## The Annals of Statistics

### General nonexact oracle inequalities for classes with a subexponential envelope

#### Abstract

We show that empirical risk minimization procedures and regularized empirical risk minimization procedures satisfy nonexact oracle inequalities in an unbounded framework, under the assumption that the class has a subexponential envelope function. The main novelty, in addition to the boundedness assumption free setup, is that those inequalities can yield fast rates even in situations in which exact oracle inequalities only hold with slower rates.

We apply these results to show that procedures based on $\ell_{1}$ and nuclear norms regularization functions satisfy oracle inequalities with a residual term that decreases like $1/n$ for every $L_{q}$-loss functions ($q\geq2$), while only assuming that the tail behavior of the input and output variables are well behaved. In particular, no RIP type of assumption or “incoherence condition” are needed to obtain fast residual terms in those setups. We also apply these results to the problems of convex aggregation and model selection.

#### Article information

Source
Ann. Statist. Volume 40, Number 2 (2012), 832-860.

Dates
First available in Project Euclid: 1 June 2012

https://projecteuclid.org/euclid.aos/1338515139

Digital Object Identifier
doi:10.1214/11-AOS965

Mathematical Reviews number (MathSciNet)
MR2933668

Zentralblatt MATH identifier
1274.62247

#### Citation

Lecué, Guillaume; Mendelson, Shahar. General nonexact oracle inequalities for classes with a subexponential envelope. Ann. Statist. 40 (2012), no. 2, 832--860. doi:10.1214/11-AOS965. https://projecteuclid.org/euclid.aos/1338515139.

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#### Supplemental materials

• Supplementary material: Applications to matrix completion, convex aggregation and model selection. In the supplementary file, we apply our main results to the problem of matrix completion, convex aggregation and model selection. The aim is to expose the fundamental differences between exact and nonexact oracle inequalities on classical problems.