General nonexact oracle inequalities for classes with a subexponential envelope

Guillaume Lecué (1, 2) , Shahar Mendelson (3)

1 UPEM - Université Paris-Est Marne-la-Vallée
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
3 Technion - Israel Institute of Technology [Haifa]

We show that empirical risk minimization procedures and regularized empirical risk minimization procedures satisfy non-exact 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 behaviour 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.

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Guillaume Lecué, Shahar Mendelson. General nonexact oracle inequalities for classes with a subexponential envelope. Annals of Statistics, 2012, 40 (2), pp.832-860. ⟨hal-00736209⟩