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Robust Wasserstein profile inference and applications to machine learning

Part of: Limit theorems Linear inference, regression

Published online by Cambridge University Press:  01 October 2019

Jose Blanchet ,
Yang Kang  and
Karthyek Murthy
Jose Blanchet*
Affiliation:
Stanford University
Yang Kang*
Affiliation:
Columbia University
Karthyek Murthy*
Affiliation:
Singapore University of Technology and Design
*
*Postal address: Management Science and Engineering, Stanford University, 475 Via Ortega, Stanford, CA 94305, USA.
**Postal address: Columbia University, 1255 Amsterdam Avenue, Rm 1005, New York, NY 10027, USA.
***Postal address: Singapore University of Technology and Design, 8 Somapah Road, Singapore 487372, Singapore.
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Abstract

We show that several machine learning estimators, including square-root least absolute shrinkage and selection and regularized logistic regression, can be represented as solutions to distributionally robust optimization problems. The associated uncertainty regions are based on suitably defined Wasserstein distances. Hence, our representations allow us to view regularization as a result of introducing an artificial adversary that perturbs the empirical distribution to account for out-of-sample effects in loss estimation. In addition, we introduce RWPI (robust Wasserstein profile inference), a novel inference methodology which extends the use of methods inspired by empirical likelihood to the setting of optimal transport costs (of which Wasserstein distances are a particular case). We use RWPI to show how to optimally select the size of uncertainty regions, and as a consequence we are able to choose regularization parameters for these machine learning estimators without the use of cross validation. Numerical experiments are also given to validate our theoretical findings.

Keywords

Distributionally robust optimizationWasserstein distanceregularizationsquare-root LASSOlogistic regressionsupport vector machinelimit characterization of optimal Wasserstein ball radius and regularization parameterempirical likelihood

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 62J05: Linear regression 62J12: Generalized linear models
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 3 , September 2019 , pp. 830 - 857
Copyright
© Applied Probability Trust 2019 

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Footnotes

The supplementary material for this article can be found at http://doi.org/10.1017/jpr.2019.49

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