In most classification tasks there are observations that are ambiguous and therefore difficult to correctly label. Set-valued classification allows the classifiers to output a set of plausible labels rather than a single label, thereby giving a more appropriate and informative treatment to the labeling of ambiguous instances. We introduce a framework for multiclass set-valued classification, where the classifiers guarantee user-defined levels of coverage or confidence (the probability that the true label is contained in the set) while minimizing the ambiguity (the expected size of the output). We first derive oracle classifiers assuming the true distribution to be known. We show that the oracle classifiers are obtained from level sets of the functions that define the conditional probability of each class. Then we develop estimators with good asymptotic and finite sample properties. The proposed classifiers build on and refine many existing single-label classifiers. The optimal classifier can sometimes output the empty set. We provide two solutions to fix this issue that are suitable for various practical needs.

此研究针对分类问题中存在的模糊不确定标签，提出了一种多类别集合值分类方法，在最小化分类的模糊性的前提下，保证用户定义的覆盖率和置信度，利用条件概率函数的级别集导出oracle分类器，并提出一种在现有单标记分类器基础上的估计器，可以在不同的实际应用需求下解决因优化而导致的输出空集问题。

具有有界误差水平的最少歧义集值分类器