We examine a stochastic formulation for data-driven optimization wherein the decision-maker is not privy to the true distribution, but has knowledge that it lies in some hypothesis set and possesses a historical data set, from which information about it can be gleaned. We define a prescriptive solution as a decision rule mapping such a data set to decisions. As there does not exist prescriptive solutions that are generalizable over the entire hypothesis set, we define out-of-sample optimality as a local average over a neighbourhood of hypotheses, and averaged over the sampling distribution. We prove sufficient conditions for local out-of-sample optimality, which reduces to functions of the sufficient statistic of the hypothesis family. We present an optimization problem that would solve for such an out-of-sample optimal solution, and does so efficiently by a combination of sampling and bisection search algorithms. Finally, we illustrate our model on the newsvendor model, and find strong performance when compared against alternatives in the literature. There are potential implications of our research on end-to-end learning and Bayesian optimization.

翻译：我们考察了一种数据驱动优化的随机规划形式，其中决策者无法获知真实分布，但已知该分布位于某个假设集中，并拥有一个历史数据集，可从中提取相关信息。我们将一种处方性解定义为将此类数据集映射到决策的决策规则。由于不存在能够在整个假设集上泛化的处方性解，我们将样本外最优性定义为假设邻域上的局部平均值，并基于采样分布取平均。我们证明了局部样本外最优性的充分条件，该条件归结为假设族充分统计量的函数。我们提出一个能求解此类样本外最优解的优化问题，并通过采样与二分搜索算法的组合高效实现。最后，我们在报童模型上展示了我们的模型，并发现与文献中的替代方案相比具有强性能。我们的研究对端到端学习和贝叶斯优化具有潜在意义。