Jun-Ya Gotoh, Michael Jong Kim, Andrew E. B. Lim
arXiv 17 Nov 2017 · Statistics — Machine Learning · publishedOperations Research (2017) · 15 citations (OpenAlex)
arXiv:1711.06565 · PDF · DOI · OpenAlex · Extracted main text
We study the out-of-sample properties of robust empirical optimization problems with smooth $\phi$-divergence penalties and smooth concave objective functions, and develop a theory for data-driven calibration of the non-negative "robustness parameter" $\delta$ that controls the size of the deviations from the nominal model. Building on the intuition that robust optimization reduces the sensitivity of the expected reward to errors in the model by controlling the spread of the reward distribution, we show that the first-order benefit of “little bit of robustness" (i.e., $\delta$ small, positive) is a significant reduction in the variance of the out-of-sample reward while the corresponding impact on the mean is almost an order of magnitude smaller. One implication is that substantial variance (sensitivity) reduction is possible at little cost if the robustness parameter is properly calibrated. To this end, we introduce the notion of a robust mean-variance frontier to select the robustness parameter and show that it can be approximated using resampling methods like the bootstrap. Our examples show that robust solutions resulting from "open loop" calibration methods (e.g., selecting a $90%$ confidence level regardless of the data and objective function) can be very conservative out-of-sample, while those corresponding to the robustness parameter that optimizes an estimate of the out-of-sample expected reward (e.g., via the bootstrap) with no regard for the variance are often insufficiently robust.
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| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Gotoh, J., Kim, M.J., Lim, A.E.B (2018) Robust empirical optimization is almost the same as mean-variance optimization self | 0.969 | 11 | 4 | 91% |
| 2 | Lam, H (2016) Robust sensitivity analysis for stochastic systems | 0.874 | 6 | 2 | 100% |
| 3 | Shapiro, A., Dentcheva, D., Ruszczyński, A (2014) Lectures on Stochastic Programming: Modeling and Theory, Second Edition | 0.855 | 8 | 3 | 62% |
| 4 | Ben-Tal, A., den Hertog, D., De Waegenaere, A., Melenberg, B., Renne… (2013) Robust solutions of optimization problems affected by uncertain probabilities | 0.843 | 4 | 4 | 75% |
| 5 | Bertsimas, D., Copenhaver, M.S (2018) Characterization of the equivalence of robustification and regularization in linear and matrix regression | 0.811 | 4 | 2 | 100% |
| 6 | Blanchet, J., Kang, Y., Zhang, F., Murthy, K (2017) Data-driven optimal cost selection for distributionally robust optimization | 0.811 | 4 | 2 | 100% |
| 7 | Gao, R., Chen, X., Kleywegt, A.J (2017) Wasserstein Distributional Robustness and Regularization in Statistical Learning | 0.811 | 4 | 2 | 100% |
| 8 | Xu, H., Caramanis, C, Mannor, S (2010) Robust Regression and Lasso | 0.811 | 4 | 2 | 100% |
| 9 | van der Vaart, A.W (2000) Asymptotic Statistics | 0.781 | 7 | 2 | 71% |
| 10 | Bertsimas, D., Litvinov, E., Sun, A.X., Zhao, J., Zheng, T (2013) Adaptive robust optimization for the security constrained unit commitment problem | 0.737 | 3 | 2 | 100% |
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arXiv econ.EM papers that cite this one, ranked by how heavily they lean on it.
| Citing paper | Intensity | Mentions | Sections | |
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| 1 | Worst-case sensitivity | 0.585 | 3 | 1 |
| 2 | Robust decision-making under risk and ambiguity | 0.405 | 1 | 1 |
| 3 | Posterior and Likelihood Sensitivity in Bayesian Distributionally Robust Optimization | 0.405 | 1 | 1 |