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Estimation bounds and sharp oracle inequalities of regularized procedures with Lipschitz loss functions

Author

Listed:
  • Pierre Alquier

    (CREST; ENSAE; Université Paris Saclay)

  • Vincent Cottet

    (CREST; ENSAE; Université Paris Saclay)

  • Guillaume Lecué

    (CREST; CNRS; Université Paris Saclay)

Abstract

We obtain estimation error rates and sharp oracle inequalities for regularization procedures of the form [See the abstract on the paper for the formula] when ||.|| is any norm, F is a convex class of functions and l is a Lipschitz loss function satisfying a Bernstein condition over F. We explore both the bounded and subgaussian stochastic frameworks for the distribution of the f(Xi)'s, with no assumption on the distribution of the Yi'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 l and norm ||.||). 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; 3) kernel methods, where the loss is the hinge loss, and the regularization function is the RKHS norm.

Suggested Citation

  • Pierre Alquier & Vincent Cottet & Guillaume Lecué, 2017. "Estimation bounds and sharp oracle inequalities of regularized procedures with Lipschitz loss functions," Working Papers 2017-30, Center for Research in Economics and Statistics.
  • Handle: RePEc:crs:wpaper:2017-30
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    References listed on IDEAS

    as
    1. Lukas Meier & Sara Van De Geer & Peter Bühlmann, 2008. "The group lasso for logistic regression," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(1), pages 53-71, February.
    2. Tian, Guo-Liang & Tang, Man-Lai & Fang, Hong-Bin & Tan, Ming, 2008. "Efficient methods for estimating constrained parameters with applications to regularized (lasso) logistic regression," Computational Statistics & Data Analysis, Elsevier, vol. 52(7), pages 3528-3542, March.
    3. Alexandre Belloni & Victor Chernozhukov, 2009. "L1-Penalized Quantile Regression in High-Dimensional Sparse Models," Papers 0904.2931, arXiv.org, revised Sep 2019.
    4. Bartlett, Peter L. & Jordan, Michael I. & McAuliffe, Jon D., 2006. "Convexity, Classification, and Risk Bounds," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 138-156, March.
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