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Characterization of the equivalence of robustification and regularization in linear and matrix regression

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  • Bertsimas, Dimitris
  • Copenhaver, Martin S.

Abstract

The notion of developing statistical methods in machine learning which are robust to adversarial perturbations in the underlying data has been the subject of increasing interest in recent years. A common feature of this work is that the adversarial robustification often corresponds exactly to regularization methods which appear as a loss function plus a penalty. In this paper we deepen and extend the understanding of the connection between robustification and regularization (as achieved by penalization) in regression problems. Specifically, (a)In the context of linear regression, we characterize precisely under which conditions on the model of uncertainty used and on the loss function penalties robustification and regularization are equivalent.(b)We extend the characterization of robustification and regularization to matrix regression problems (matrix completion and Principal Component Analysis).

Suggested Citation

  • Bertsimas, Dimitris & Copenhaver, Martin S., 2018. "Characterization of the equivalence of robustification and regularization in linear and matrix regression," European Journal of Operational Research, Elsevier, vol. 270(3), pages 931-942.
    • Handle: RePEc:eee:ejores:v:270:y:2018:i:3:p:931-942
    DOI: 10.1016/j.ejor.2017.03.051
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    Citations

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    Cited by:

    1. Bottmer, Lea & Croux, Christophe & Wilms, Ines, 2022. "Sparse regression for large data sets with outliers," European Journal of Operational Research, Elsevier, vol. 297(2), pages 782-794.
    2. Bertsimas, Dimitris & Koukouvinos, Thodoris, 2024. "Robust linear algebra," European Journal of Operational Research, Elsevier, vol. 314(3), pages 1174-1184.
    3. Jun-ya Gotoh & Michael Jong Kim & Andrew E. B. Lim, 2020. "Worst-case sensitivity," Papers 2010.10794, arXiv.org.
    4. Joscha Krause & Jan Pablo Burgard & Domingo Morales, 2022. "Robust prediction of domain compositions from uncertain data using isometric logratio transformations in a penalized multivariate Fay–Herriot model," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 76(1), pages 65-96, February.
    5. Mengshi Lu & Zuo‐Jun Max Shen, 2021. "A Review of Robust Operations Management under Model Uncertainty," Production and Operations Management, Production and Operations Management Society, vol. 30(6), pages 1927-1943, June.
    6. Jan Pablo Burgard & Joscha Krause & Ralf Münnich, 2019. "Penalized Small Area Models for the Combination of Unit- and Area-level Data," Research Papers in Economics 2019-05, University of Trier, Department of Economics.
    7. Jan Pablo Burgard & Joscha Krause & Dennis Kreber, 2019. "Regularized Area-level Modelling for Robust Small Area Estimation in the Presence of Unknown Covariate Measurement Errors," Research Papers in Economics 2019-04, University of Trier, Department of Economics.
    8. Jan Pablo Burgard & Joscha Krause & Dennis Kreber & Domingo Morales, 2021. "The generalized equivalence of regularization and min–max robustification in linear mixed models," Statistical Papers, Springer, vol. 62(6), pages 2857-2883, December.
    9. Carina Moreira Costa & Dennis Kreber & Martin Schmidt, 2022. "An Alternating Method for Cardinality-Constrained Optimization: A Computational Study for the Best Subset Selection and Sparse Portfolio Problems," INFORMS Journal on Computing, INFORMS, vol. 34(6), pages 2968-2988, November.
    10. Jack Dunn & Ying Daisy Zhuo, 2022. "Detecting Racial Bias in Jury Selection," SN Operations Research Forum, Springer, vol. 3(3), pages 1-17, September.
    11. Gambella, Claudio & Ghaddar, Bissan & Naoum-Sawaya, Joe, 2021. "Optimization problems for machine learning: A survey," European Journal of Operational Research, Elsevier, vol. 290(3), pages 807-828.

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