## Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

### Sparsity in penalized empirical risk minimization

#### Abstract

Let $(X, Y)$ be a random couple in $S×T$ 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 $h_1, …, h_N:S↦[−1, 1]$ be a dictionary consisting of $N$ functions. For $λ∈ℝ^N$, denote $f_λ:=∑_{j=1}^Nλ_jh_j$. Let $ℓ:T×ℝ↦ℝ$ 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 $$\hat{\lambda}^{\varepsilon }:=\mathop{\operatorname {argmin}}_{\lambda\in {\mathbb{R}}^{N}}\bigl[P_{n}(\ell\bullet f_{\lambda})+\varepsilon \|\lambda\|_{\ell_{p}}^{p}\bigr],$$ which is an empirical version of the problem $$\lambda^{\varepsilon }:=\mathop{\operatorname{argmin}}_{\lambda\in {\mathbb{R}}^{N}}\bigl[P(\ell \bullet f_{\lambda})+\varepsilon \|\lambda\|_{\ell_{p}}^{p}\bigr]$$ (here $\varepsilon≥0$ is a regularization parameter; $λ^0$ corresponds to $\varepsilon=0$). A number of regression and classification problems fit this general framework. We are interested in the case when $p≥1$, 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 $λ^\varepsilon$ implies the “sparsity” of $\hat{\lambda}^\varepsilon$ and study the impact of “sparsity” on bounding the excess risk $P(ℓ•f_{{\hat{\lambda}^\varepsilon}})−P(ℓ•f_{{λ^0}})$ of solutions of empirical risk minimization problems.

#### Résumé

Soit $(X, Y)$ un couple aléatoire à valeurs dans $S×T$ 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 $h_1, …, h_N:S↦[−1, 1]$ un dictionnaire composé de $N$ fonctions. Pour tout $λ∈ℝ^N$, on note $f_λ:=∑_{j=1}^Nλ_jh_j$. Soit $ℓ:T×ℝ↦ℝ$ 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 $$\hat{\lambda}^{\varepsilon }:=\mathop{\operatorname {argmin}}_{\lambda\in {\mathbb{R}}^{N}}\bigl[P_{n}(\ell\bullet f_{\lambda})+\varepsilon \|\lambda\|_{\ell_{p}}^{p}\bigr],$$ qui correspond à la version empirique du problème $$\lambda^{\varepsilon }:=\mathop{\operatorname{argmin}}_{\lambda\in {\mathbb{R}}^{N}}\bigl[P(\ell \bullet f_{\lambda})+\varepsilon \|\lambda\|_{\ell_{p}}^{p}\bigr]$$ (ici $\varepsilon≥0$ est un paramètre de régularisation; $λ^0$ correspond au cas $\varepsilon=0$). Ce cadre général englobe un certain nombre de problèmes de régression et de classification. On s’intéresse au cas où $p≥1$, 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 $λ^\varepsilon$ implique la “sparsité” de $\hat{\lambda}^\varepsilon$. En outre, on étudie les conséquences de la “sparsité” en termes de bornes supérieures sur l’excès de risque $P(ℓ•f_{\hat{\lambda}^\varepsilon})−P(ℓ•f_{λ^0})$ des solutions obtenues pour les différents problèmes de minimisation du risque empirique.

#### Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 45, Number 1 (2009), 7-57.

Dates
First available in Project Euclid: 12 February 2009

https://projecteuclid.org/euclid.aihp/1234469970

Digital Object Identifier
doi:10.1214/07-AIHP146

Mathematical Reviews number (MathSciNet)
MR2500227

Zentralblatt MATH identifier
1168.62044

#### Citation

Koltchinskii, Vladimir. Sparsity in penalized empirical risk minimization. Ann. Inst. H. Poincaré Probab. Statist. 45 (2009), no. 1, 7--57. doi:10.1214/07-AIHP146. https://projecteuclid.org/euclid.aihp/1234469970

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