From Predictive to Prescriptive Analytics

发表在 Management Science, 2020. DOI: https://doi.org/10.1287/mnsc.2018.3253.

Area of review: optimization.

Keywords: data-driven decision making; machine learning; stochastic optimization

We combine ideas from machine learning (ML) and operations research and management science (OR/MS) in developing a framework, along with speciﬁc methods, for using data to prescribe optimal decisions in OR/MS problems.

We demonstrate how our proposed methods are generally applicable to a wide range of decision problems and prove that they are computationally tractable and asymptotically optimal under mild conditions, even when data are not independent and identically distributed and for censored observations.

这篇文章写于2015年，

很多管理学问题都包含以下3种元素：

需要预测的数据： ${y^{1}, \dots, y^{N}}$ ，比如需求
特征数据： ${x^{1}, \dots, x^{N}}$
决策 $z$ ，在观测到特征 $X$ 后做出，决策的目标是最小化损失函数 $c (z; Y)$

传统的决策方法是： $\begin{aligned} v^{stoch} = min_{z \in Z} E [c (z; Y)], \\ z^{stoch} \in \underset{z \in Z}{\arg min} E [c (z; Y)] \end{aligned}$ 这并没有把数据考虑进去。

与此同时，蓬勃发展的机器学习极大地促进了预测 (prediction) 的发展，而这篇文章想解决的问题是，如何从做出好的预测转变为做出好的决策？

文章试图解决： $\begin{aligned} v^{*} (x) = min_{z \in L} E [c (z; Y) ∣ X = x], \\ z^{*} (x) \in L^{*} (x) = \underset{z \in Z}{\arg min} E [c (z; Y) ∣ X = x], \end{aligned}$ 其中 $(X, Y)$ 的联合分布 $μ_{X, Y}$ 是未知的，我们只有数据 $S_{N} = {(x^{1}, y^{1}), \dots, (x^{N}, y^{N})}$ .

From Data to Predictive Prescriptions

我们希望在观测到特征 $X = x$ 后做出决策 $z$ 来最小化 $c (z; Y)$ ，但是我们只有数据 $S^{N}$ .

文章提出的规范性方法都具有如下的形式： ${\hat{z}}_{N}^{local} (x) \in \underset{z \in Z}{\arg min} \sum_{i = 1}^{N} w_{N, i} (x) c (z; y^{i})$

KNN

$w_{N, i}^{k N N} (x) = \frac{1}{k} I [x^{i} is a k N N of x]$

Kernel Methods

也可以用核函数来估计 $x$ 的距离 $w_{N, i}^{K R} (x) = \frac{K ((x^{i} - x) / h_{N})}{\sum_{j = 1}^{N} K ((x^{j} - x) / h_{N})}$

From Predictive to Prescriptive Analytics

From Data to Predictive Prescriptions

KNN

Kernel Methods

Local Linear Methods

Trees

Ensembles

From Data to Predictive Prescriptions When Decisions Affect Uncertainty

From Predictive to Prescriptive Analytics

From Data to Predictive Prescriptions

KNN

Kernel Methods

Local Linear Methods

Trees

Ensembles

From Data to Predictive Prescriptions When Decisions Affect Uncertainty

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