Jun-ya Gotoh, Michael Jong Kim, Andrew E. B. Lim
arXiv 21 Oct 2020 · Econometrics
arXiv:2010.10794 · PDF · DOI · OpenAlex · Extracted main text
We introduce the notion of Worst-Case Sensitivity, defined as the worst-case rate of increase in the expected cost of a Distributionally Robust Optimization (DRO) model when the size of the uncertainty set vanishes. We show that worst-case sensitivity is a Generalized Measure of Deviation and that a large class of DRO models are essentially mean-(worst-case) sensitivity problems when uncertainty sets are small, unifying recent results on the relationship between DRO and regularized empirical optimization with worst-case sensitivity playing the role of the regularizer. More generally, DRO solutions can be sensitive to the family and size of the uncertainty set, and reflect the properties of its worst-case sensitivity. We derive closed-form expressions of worst-case sensitivity for well known uncertainty sets including smooth $\phi$-divergence, total variation, "budgeted" uncertainty sets, uncertainty sets corresponding to a convex combination of expected value and CVaR, and the Wasserstein metric. These can be used to select the uncertainty set and its size for a given application.
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| Reference | Intensity | Mentions | Sections | Main text | |
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| 1 | Jun-ya Gotoh, Michael Jong Kim, and Andrew E.B. Lim (2018) Robust empirical optimization is almost the same as mean–variance optimization self | 0.950 | 7 | 4 | 86% |
| 2 | R. Tyrrell Rockafellar, Stan Uryasev, and Michael Zabarankin (2006) Generalized deviations in risk analysis | 0.928 | 4 | 4 | 100% |
| 3 | Jose Blanchet, Yang Kang, and Karthyek Murthy (2019) Robust Wasserstein profile inference and applications to machine learning | 0.811 | 4 | 2 | 100% |
| 4 | Rui Gao, Xi Chen, and Anton J Kleywegt (2017) Wasserstein distributional robustness and regularization in statistical learning | 0.811 | 4 | 2 | 100% |
| 5 | David G. Luenberger (1997) Optimization by Vector Space Methods | 0.737 | 5 | 2 | 60% |
| 6 | Soroosh Shafieezadeh-Abadeh, Peyman Mohajerin Esfahani, and Daniel K… (2015) Distributionally robust logistic regression | 0.737 | 4 | 2 | 75% |
| 7 | Henry Lam (2016) Robust sensitivity analysis for stochastic systems | 0.644 | 2 | 2 | 100% |
| 8 | Jun-ya Gotoh, Michael Jong Kim, and Andrew E.B. Lim (2020) Calibration of distributionally robust empirical optimization models self | 0.585 | 3 | 1 | 100% |
| 9 | John C. Duchi and Hongseok Namkoong (2019) Variance-based regularization with convex objectives | 0.511 | 2 | 1 | 100% |
| 10 | Huan Xu, Constantine Caramanis, and Shie Mannor (2010) Robust regression and Lasso | 0.511 | 2 | 1 | 100% |
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