The entropic risk measure is widely used in high-stakes decision making to account for tail risks associated with an uncertain loss. With limited data, the empirical entropic risk estimator, i.e. replacing the expectation in the entropic risk measure with a sample average, underestimates the true risk. To mitigate the bias in the empirical entropic risk estimator, we propose a strongly asymptotically consistent bootstrapping procedure. The first step of the procedure involves fitting a distribution to the data, whereas the second step estimates the bias of the empirical entropic risk estimator using bootstrapping, and corrects for it. Two methods are proposed to fit a Gaussian Mixture Model to the data, a computationally intensive one that fits the distribution of empirical entropic risk, and a simpler one with a component that fits the tail of the empirical distribution. As an application of our approach, we study distributionally robust entropic risk minimization problems with type-$\infty$ Wasserstein ambiguity set and apply our bias correction to debias validation performance. Furthermore, we propose a distributionally robust optimization model for an insurance contract design problem that takes into account the correlations of losses across households. We show that choosing regularization parameters based on the cross validation methods can result in significantly higher out-of-sample risk for the insurer if the bias in validation performance is not corrected for. This improvement in performance can be explained from the observation that our methods suggest a higher (and more accurate) premium to homeowners.

Recently there has been a surge of interest in operations research (OR) and the machine learning (ML) community in combining prediction algorithms and optimization techniques to solve decision-making problems in the face of uncertainty. This gave rise to the field of contextual optimization, under which data-driven procedures are developed to prescribe actions to the decision-maker that make the best use of the most recently updated information. A large variety of models and methods have been presented in both OR and ML literature under a variety of names, including data-driven optimization, prescriptive optimization, predictive stochastic programming, policy optimization, (smart) predict/estimate-then-optimize, decision-focused learning, (task-based) end-to-end learning/forecasting/optimization, etc. Focusing on single and two-stage stochastic programming problems, this review article identifies three main frameworks for learning policies from data and discusses their strengths and limitations. We present the existing models and methods under a uniform notation and terminology and classify them according to the three main frameworks identified. Our objective with this survey is to both strengthen the general understanding of this active field of research and stimulate further theoretical and algorithmic advancements in integrating ML and stochastic programming.