具有协变量信息的数据驱动样本平均值逼近
Data-Driven Sample Average Approximation with Covariate Information
摘要：
我们研究数据驱动的决策优化时，我们有观察的不确定参数的最优化模型与并发观察的协变量。给定一个新的协变量观测值，目标是选择一个决策，使该观测值的预期成本最小化。我们研究了三个数据驱动的框架，这些框架将机器学习预测模型集成在随机规划样本平均近似（SAA）中，以近似解决这个问题。SAA框架中有两个是新的，使用留一预测模型的样本外残差来生成场景。我们调查的框架是灵活的，并适应参数，非参数和半参数回归技术。我们推导出的数据生成过程中，预测模型，和随机程序的条件下，这些数据驱动的SAA的解决方案是一致的，渐近最优的，也得到收敛速度和有限样本的保证。计算实验验证了我们的理论结果，证明了我们的数据驱动的配方比现有的方法（即使预测模型是错误的）的潜在优势，并说明了我们的新的数据驱动的配方在有限的数据制度的好处。
英文摘要：
We study optimization for data-driven decision-making when we have observations of the uncertain parameters within the optimization model together with concurrent observations of covariates. Given a new covariate observation, the goal is to choose a decision that minimizes the expected cost conditioned on this observation. We investigate three data-driven frameworks that integrate a machine learning prediction model within a stochastic programming sample average approximation (SAA) for approximating the solution to this problem. Two of the SAA frameworks are new and use out-of-sample residuals of leave-one-out prediction models for scenario generation. The frameworks we investigate are flexible and accommodate parametric, nonparametric, and semiparametric regression techniques. We derive conditions on the data generation process, the prediction model, and the stochastic program under which solutions of these data-driven SAAs are consistent and asymptotically optimal, and also derive convergence rates and finite sample guarantees. Computational experiments validate our theoretical results, demonstrate the potential advantages of our data-driven formulations over existing approaches (even when the prediction model is misspecified), and illustrate the benefits of our new data-driven formulations in the limited data regime.
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