We develop the first active learning method in the predict-then-optimize framework. Specifically, we develop a learning method that sequentially decides whether to request the "labels" of feature samples from an unlabeled data stream, where the labels correspond to the parameters of an optimization model for decision-making. Our active learning method is the first to be directly informed by the decision error induced by the predicted parameters, which is referred to as the Smart Predict-then-Optimize (SPO) loss. Motivated by the structure of the SPO loss, our algorithm adopts a margin-based criterion utilizing the concept of distance to degeneracy and minimizes a tractable surrogate of the SPO loss on the collected data. In particular, we develop an efficient active learning algorithm with both hard and soft rejection variants, each with theoretical excess risk (i.e., generalization) guarantees. We further derive bounds on the label complexity, which refers to the number of samples whose labels are acquired to achieve a desired small level of SPO risk. Under some natural low-noise conditions, we show that these bounds can be better than the naive supervised learning approach that labels all samples. Furthermore, when using the SPO+ loss function, a specialized surrogate of the SPO loss, we derive a significantly smaller label complexity under separability conditions. We also present numerical evidence showing the practical value of our proposed algorithms in the settings of personalized pricing and the shortest path problem.

翻译：我们提出了预测而后优化框架下的首个主动学习方法。具体而言，我们开发了一种学习方法，能够顺序决定是否从无标签数据流中请求特征样本的“标签”，其中标签对应于用于决策的优化模型参数。我们的主动学习方法是首个直接由预测参数所导致的决策误差（即智能预测而后优化损失，SPO损失）驱动的学习方法。受SPO损失结构启发，该算法采用基于边际的准则，利用退化距离的概念，并在收集的数据上最小化SPO损失的可处理替代函数。特别地，我们开发了一种高效的主动学习算法，包含硬拒绝和软拒绝两种变体，每种变体均具有理论上的超额风险（即泛化能力）保证。我们进一步推导了标签复杂度的上界，即为实现期望的小规模SPO风险而需要获取标签的样本数量。在若干自然的低噪声条件下，我们证明这些上界优于对所有样本标注的朴素监督学习方法。此外，当使用SPO+损失函数（SPO损失的一种专用替代函数）时，我们在可分性条件下推导出显著更小的标签复杂度。我们通过个性化定价和最短路径问题的数值实验，展示了所提出算法的实际应用价值。