Calibration of Distributionally Robust Empirical Optimization Models

• 成果类型: Article
• 署名作者: Gotoh, Jun-ya; Kim, Michael Jong; Lim, Andrew E. B.
• 署名单位: Chuo University; University of British Columbia; National University of Singapore; National University of Singapore; National University of Singapore
• 刊物名称: OPERATIONS RESEARCH
• ISSN/ISSBN: 0030-364X
• DOI: 10.1287/opre.2020.2041
• 发表日期: 2021
• 页码: 1630-1650

摘要

We study the out-of-sample properties of robust empirical optimization problems with smooth phi-divergence penalties and smooth concave objective functions, and we develop a theory for data-driven calibration of the nonnegative 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, whereas 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 such as 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, whereas 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.