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Sparsity in penalized empirical risk minimization. (English) Zbl 1168.62044
Summary: Let $$(X,Y)$$ be a random couple in $$S\times T$$ with unknown distribution $$P$$. Let $$(X_1,Y_1),\dots, (X_n,Y_n)$$ be i.i.d. copies of $$(X,Y)$$, $$P_n$$ being their empirical distribution. Let $$h_1,\dots, h_N:S\mapsto[-1,1]$$ be a dictionary consisting of $$N$$ functions. For $$\lambda\in\mathbb R^N$$, denote $$f_\lambda:=\sum_{j=1}^N \lambda_jh_j$$. Let $$\ell:T\times\mathbb R\mapsto\mathbb R$$ be a given loss function, which is convex with respect to the second variable. Denote $$(\ell\bullet f)(x,y):= \ell(y;f(x))$$. We study the following penalized empirical risk minimization problem: $\widehat{\lambda}^\varepsilon:= \mathop{\operatorname {argmin}}_{\lambda\in\mathbb R^N}\bigl[P_n(\ell\bullet f_\lambda)+ \varepsilon \|\lambda\|_{\ell_p}^p\bigr],$ 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 $$\varepsilon\geq0$$ is a regularization parameter; $$\lambda^0$$ corresponds to $$\varepsilon=0$$). A number of regression and classification problems fit this general framework. We are interested in the case when $$p\geq1$$, but it is close enough to 1 (so that $$p-1$$ is of the order $$1/\log N$$, or smaller). We show that the “sparsity” of $$\lambda^\varepsilon$$ implies the “sparsity” of $$\widehat{\lambda}^\varepsilon$$ and study the impact of “sparsity” on bounding the excess risk $$P(\ell\bullet f_{\widehat{\lambda}^\varepsilon})-P(\ell\bullet f_{\lambda^0})$$ of solutions of empirical risk minimization problems.

##### MSC:
 62G30 Order statistics; empirical distribution functions 62G99 Nonparametric inference 60E15 Inequalities; stochastic orderings 62J99 Linear inference, regression 62H30 Classification and discrimination; cluster analysis (statistical aspects)
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