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Some recent works showed that several machine learning algorithms, such as square-root Lasso, Support Vector Machines, and regularized logistic regression, among many others, can be represented exactly as distributionally robust optimization (DRO) problems. The distributional uncertainty set is defined as a neighborhood centered at the empirical distribution, and the neighborhood is measured by optimal transport distance. In this paper, we propose a methodology which learns such neighborhood in a natural data-driven way. We show rigorously that our framework encompasses adaptive regularization as a particular case. Moreover, we demonstrate empirically that our proposed methodology is able to improve upon a wide range of popular machine learning estimators.
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Robust Wasserstein profile inference and applications to machine learning
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.
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- Award ID(s):
- 1820942
- PAR ID:
- 10175458
- Date Published:
- Journal Name:
- Journal of Applied Probability
- Volume:
- 56
- Issue:
- 03
- ISSN:
- 0021-9002
- Page Range / eLocation ID:
- 830 to 857
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jpr.2019.49
