基于数据的学习与决策：最优公式化与相变
Learning and Decision-Making with Data: Optimal Formulations and Phase Transitions
AI总结：
该研究针对仅存历史数据的场景，先定义数据驱动公式化方案的质量基准，再构造出同样本外性能下最贴近真实成本的最优方案，发现其性质在三类性能区间间存在相变，揭示了三类看似无关的数据驱动方案的内在关联。
摘要：
我们研究了仅存在历史数据时设计最优学习与决策公式化方案的问题。已有研究通常先选定某一类数据驱动公式化方案，再尝试为其建立样本外性能保证。本文采用相反的研究路径：首先定义一个用于衡量任意数据驱动公式化方案质量的合理基准，随后寻找满足该基准的最优方案。宽泛而言，任何数据驱动公式化方案都需要在估计成本与真实成本的接近程度，以及样本外性能水平保证之间取得平衡。在给定可接受的样本外性能水平的前提下，我们显式构造了一种数据驱动公式化方案，其与真实成本的一致接近程度优于所有具备相同样本外性能的其他方案。我们证明存在三种截然不同的样本外性能区间（超指数区间、指数区间和次指数区间），最优数据驱动公式化方案的性质会在这些区间之间发生相变。这些最优数据驱动公式化方案可分别解释为：超指数区间下的经典鲁棒公式化方案、指数区间下的熵分布鲁棒公式化方案，以及次指数区间下的方差惩罚公式化方案。这一最终发现揭示了三类乍看之下毫无关联的数据驱动公式化方案之间此前未被发现的惊人关联。
英文摘要：
We study the problem of designing optimal learning and decision-making formulations when only historical data is available. Prior work typically commits to a particular class of data-driven formulation and subsequently tries to establish out-of-sample performance guarantees. We take here the opposite approach. We define first a sensible yard stick with which to measure the quality of any data-driven formulation and subsequently seek to find an optimal such formulation. Informally, any data-driven formulation can be seen to balance a measure of proximity of the estimated cost to the actual cost while guaranteeing a level of out-of-sample performance. Given an acceptable level of out-of-sample performance, we construct explicitly a data-driven formulation that is uniformly closer to the true cost than any other formulation enjoying the same out-of-sample performance. We show the existence of three distinct out-of-sample performance regimes (a superexponential regime, an exponential regime and a subexponential regime) between which the nature of the optimal data-driven formulation experiences a phase transition. The optimal data-driven formulations can be interpreted as a classically robust formulation in the superexponential regime, an entropic distributionally robust formulation in the exponential regime and finally a variance penalized formulation in the subexponential regime. This final observation unveils a surprising connection between these three, at first glance seemingly unrelated, data-driven formulations which until now remained hidden.
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