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Learning from MOM’s principles: Le Cam’s approach

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  • Lecué, Guillaume
  • Lerasle, Matthieu

Abstract

New robust estimators are introduced, derived from median-of-means principle and Le Cam’s aggregation of tests. Minimax sparse rates of convergence are obtained with exponential probability, under weak moment’s assumptions and possible contamination of the dataset. These derive from general risk bounds of the following informal structure maxminimax rate in the i.i.d. setup,number of outliersnumber of observations.In this result, the number of outliers may be as large as (number of data)×(minimax rate) without affecting the rates. As an example, minimax rates slog(ed∕s)∕N of recovery of s-sparse vectors in Rd holding with exponentially large probability, are deduced for median-of-means versions of the LASSO when the noise has q0 moments for some q0>2, the entries of the design matrix have C0log(ed) moments and the dataset is corrupted by up to C1slog(ed∕s) outliers.

Suggested Citation

  • Lecué, Guillaume & Lerasle, Matthieu, 2019. "Learning from MOM’s principles: Le Cam’s approach," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4385-4410.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:11:p:4385-4410
    DOI: 10.1016/j.spa.2018.11.024
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    References listed on IDEAS

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    1. Baraud, Y. & Birgé, L., 2016. "Rho-estimators for shape restricted density estimation," Stochastic Processes and their Applications, Elsevier, vol. 126(12), pages 3888-3912.
    2. Jianqing Fan & Quefeng Li & Yuyan Wang, 2017. "Estimation of high dimensional mean regression in the absence of symmetry and light tail assumptions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(1), pages 247-265, January.
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    Cited by:

    1. P Alquier & M Gerber, 2024. "Universal robust regression via maximum mean discrepancy," Biometrika, Biometrika Trust, vol. 111(1), pages 71-92.
    2. Brunet-Saumard, Camille & Genetay, Edouard & Saumard, Adrien, 2022. "K-bMOM: A robust Lloyd-type clustering algorithm based on bootstrap median-of-means," Computational Statistics & Data Analysis, Elsevier, vol. 167(C).

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