Sparsity in penalized empirical risk minimization
Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 1, pp. 7-57.

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 N , denote f λ := j=1 N λ j h j . Let :T× be a given loss function, which is convex with respect to the second variable. Denote (f)(x,y):=(y;f(x)). We study the following penalized empirical risk minimization problem

λ ^ ε :=argmin λ N P n (f λ ) + ε λ p p ,
which is an empirical version of the problem
λ ε :=argmin λ N P (f λ ) + ε λ p p
(here ε0 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 p1, but it is close enough to 1 (so that p-1 is of the order 1 logN, or smaller). We show that the “sparsity” of λ ε implies the “sparsity” of λ ^ ε and study the impact of “sparsity” on bounding the excess risk P(f λ ^ ε )-P(f λ 0 ) of solutions of empirical risk minimization problems.

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 N , on note f λ := j=1 N λ j h j . Soit :T× fonction de perte donnée que l’on suppose convexe en la seconde variable. On note (f)(x,y):=(y;f(x)). On étudie le problème de minimisation du risque empirique pénalisé suivant

λ ^ ε :=argmin λ N P n (f λ ) + ε λ p p ,
qui correspond à la version empirique du problème
λ ε :=argmin λ N P (f λ ) + ε λ p p
(ici ε0 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 p1, mais reste proche de 1 (de sorte que p-1 soit de l’ordre 1 logN, ou inférieur). On montre que la «sparsité» de λ ε implique la «sparsité» de λ ^ ε . En outre, on étudie les conséquences de la «sparsité» en termes de bornes supérieures sur l’excès de risque P(f λ ^ ε )-P(f λ 0 ) des solutions obtenues pour les différents problèmes de minimisation du risque empirique.

DOI: 10.1214/07-AIHP146
Classification: 62G99,  62J99,  62H30
Keywords: empirical risk, penalized empirical risk, ℓ_p-penalty, sparsity, oracle inequalities
     author = {Koltchinskii, Vladimir},
     title = {Sparsity in penalized empirical risk minimization},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {7--57},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {1},
     year = {2009},
     doi = {10.1214/07-AIHP146},
     zbl = {1168.62044},
     mrnumber = {2500227},
     language = {en},
     url = {}
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%J Annales de l'I.H.P. Probabilités et statistiques
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Koltchinskii, Vladimir. Sparsity in penalized empirical risk minimization. Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 1, pp. 7-57. doi : 10.1214/07-AIHP146.

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