Calibration of Distributionally Robust Empirical Optimization Models

TL;DR

This paper develops a data-driven method to calibrate the robustness parameter in distributionally robust optimization, significantly reducing variance with minimal impact on mean reward, and introduces a robust mean-variance frontier for optimal calibration.

Contribution

It introduces a theory for calibrating the robustness parameter using resampling methods, balancing variance reduction and mean reward impact in robust optimization.

Findings

Robust optimization reduces reward variance significantly with minimal mean impact.

Calibrating the robustness parameter via bootstrap improves out-of-sample performance.

Open loop calibration methods can be overly conservative or insufficiently robust.

Abstract

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.