Sparsity in penalized empirical risk minimization

Koltchinskii, Vladimir

Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009), p. 7-57 / Harvested from Numdam

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Abstract

Soit $(X, Y)$ un couple aléatoire à valeurs dans $S \times T$ et de loi $P$ inconnue. Soient $(X_{1}, Y_{1}), ..., (X_{n}, Y_{n})$ des répliques i.i.d. de $(X, Y)$ , de loi empirique associée $P_{n}$ . Soit $h_{1}, ..., h_{N} : S \mapsto [- 1, 1]$ un dictionnaire composé de $N$ fonctions. Pour tout $λ \in ℝ^{N}$ , on note $f_{λ} : = \sum_{j = 1}^{N} λ_{j} h_{j}$ . Soit $ℓ : T \times ℝ \mapsto ℝ$ fonction de perte donnée que l’on suppose convexe en la seconde variable. On note $(ℓ • f) (x, y) : = ℓ (y; f (x))$ . On étudie le problème de minimisation du risque empirique pénalisé suivant ${\hat{λ}}^{ε} : = \underset{λ \in ℝ^{N}}{argmin} [P_{n} (ℓ • f_{λ}) + ε {∥ λ ∥}_{ℓ_{p}}^{p}],$ qui correspond à la version empirique du problème $λ^{ε} : = \underset{λ \in ℝ^{N}}{argmin} [P (ℓ • f_{λ}) + ε {∥ λ ∥}_{ℓ_{p}}^{p}]$ (ici $ε \geq 0$ est un paramètre de régularisation; $λ^{0}$ correspond au cas $ε = 0$ ). Ce cadre général englobe un certain nombre de problèmes de régression et de classification. On s’intéresse au cas où $p \geq 1$ , mais reste proche de 1 (de sorte que $p - 1$ soit de l’ordre $\frac{1}{log N}$ , ou inférieur). On montre que la «sparsité» de $λ^{ε}$ implique la «sparsité» de ${\hat{λ}}^{ε}$ . En outre, on étudie les conséquences de la «sparsité» en termes de bornes supérieures sur l’excès de risque $P (ℓ • f_{{\hat{λ}}^{ε}}) - P (ℓ • f_{λ^{0}})$ des solutions obtenues pour les différents problèmes de minimisation du risque empirique.

Let $(X, Y)$ be a random couple in $S \times T$ with unknown distribution $P$ . Let $(X_{1}, Y_{1}), ..., (X_{n}, Y_{n})$ be i.i.d. copies of $(X, Y)$ , $P_{n}$ being their empirical distribution. Let $h_{1}, ..., h_{N} : S \mapsto [- 1, 1]$ be a dictionary consisting of $N$ functions. For $λ \in ℝ^{N}$ , denote $f_{λ} : = \sum_{j = 1}^{N} λ_{j} h_{j}$ . Let $ℓ : T \times ℝ \mapsto ℝ$ be a given loss function, which is convex with respect to the second variable. Denote $(ℓ • f) (x, y) : = ℓ (y; f (x))$ . We study the following penalized empirical risk minimization problem ${\hat{λ}}^{ε} : = \underset{λ \in ℝ^{N}}{argmin} [P_{n} (ℓ • f_{λ}) + ε {∥ λ ∥}_{ℓ_{p}}^{p}],$ which is an empirical version of the problem $λ^{ε} : = \underset{λ \in ℝ^{N}}{argmin} [P (ℓ • f_{λ}) + ε {∥ λ ∥}_{ℓ_{p}}^{p}]$ (here $ε \geq 0$ is a regularization parameter; $λ^{0}$ corresponds to $ε = 0$ ). A number of regression and classification problems fit this general framework. We are interested in the case when $p \geq 1$ , but it is close enough to 1 (so that $p - 1$ is of the order $\frac{1}{log N}$ , or smaller). We show that the “sparsity” of $λ^{ε}$ implies the “sparsity” of ${\hat{λ}}^{ε}$ and study the impact of “sparsity” on bounding the excess risk $P (ℓ • f_{{\hat{λ}}^{ε}}) - P (ℓ • f_{λ^{0}})$ of solutions of empirical risk minimization problems.

Publié le : 2009-01-01

MR 2500227 | Zbl 1168.62044 | 1 citation dans Numdam

DOI : https://doi.org/10.1214/07-AIHP146
Classification: 62G99, 62J99, 62H30

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     author = {Koltchinskii, Vladimir},
     title = {Sparsity in penalized empirical risk minimization},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {45},
     year = {2009},
     pages = {7-57},
     doi = {10.1214/07-AIHP146},
     mrnumber = {2500227},
     zbl = {1168.62044},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2009__45_1_7_0}
}

Koltchinskii, Vladimir. Sparsity in penalized empirical risk minimization. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) pp. 7-57. doi : 10.1214/07-AIHP146. http://gdmltest.u-ga.fr/item/AIHPB_2009__45_1_7_0/

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