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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.

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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