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• W2598989315 abstract "We obtain estimation error rates and sharp oracle inequalities for regularization procedures of the form begin{equation*}hat{f}inmathop{operatorname{argmin}}_{fin F}Bigg(frac{1}{N}sum_{i=1}^{N}ell_{f}(X_{i},Y_{i})+lambda Vert fVert Bigg)end{equation*} when $Vert cdot Vert $ is any norm, $F$ is a convex class of functions and $ell$ is a Lipschitz loss function satisfying a Bernstein condition over $F$. We explore both the bounded and sub-Gaussian stochastic frameworks for the distribution of the $f(X_{i})$’s, with no assumption on the distribution of the $Y_{i}$’s. The general results rely on two main objects: a complexity function and a sparsity equation, that depend on the specific setting in hand (loss $ell$ and norm $Vert cdot Vert $). As a proof of concept, we obtain minimax rates of convergence in the following problems: (1) matrix completion with any Lipschitz loss function, including the hinge and logistic loss for the so-called 1-bit matrix completion instance of the problem, and quantile losses for the general case, which enables to estimate any quantile on the entries of the matrix; (2) logistic LASSO and variants such as the logistic SLOPE, and also shape constrained logistic regression; (3) kernel methods, where the loss is the hinge loss, and the regularization function is the RKHS norm." @default.
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• W2598989315 date "2019-08-01" @default.
• W2598989315 modified "2023-10-16" @default.
• W2598989315 title "Estimation bounds and sharp oracle inequalities of regularized procedures with Lipschitz loss functions" @default.
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• W2598989315 doi "https://doi.org/10.1214/18-aos1742" @default.
• W2598989315 hasPublicationYear "2019" @default.
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