In a classification task, counterfactual explanations provide the minimum change needed for an input to be classified into a favorable class. We consider the problem of privately retrieving the exact closest counterfactual from a database of accepted samples while enforcing that certain features of the input sample cannot be changed, i.e., they are \emph{immutable}. An applicant (user) whose feature vector is rejected by a machine learning model wants to retrieve the sample closest to them in the database without altering a private subset of their features, which constitutes the immutable set. While doing this, the user should keep their feature vector, immutable set and the resulting counterfactual index information-theoretically private from the institution. We refer to this as immutable private counterfactual retrieval (I-PCR) problem which generalizes PCR to a more practical setting. In this paper, we propose two I-PCR schemes by leveraging techniques from private information retrieval (PIR) and characterize their communication costs. Further, we quantify the information that the user learns about the database and compare it for the proposed schemes.

翻译：在分类任务中，反事实解释提供了将输入分类到有利类别所需的最小改变。我们考虑从已接受样本数据库中私有检索确切最近反事实的问题，同时强制要求输入样本的某些特征不可更改，即它们是\emph{不可变的}。一个特征向量被机器学习模型拒绝的申请者（用户）希望从数据库中检索与其最接近的样本，且不改变其特征的私有子集（即不可变集合）。在此过程中，用户应使其特征向量、不可变集合以及最终的反事实索引对机构保持信息论意义上的隐私。我们将此称为不可变私有反事实检索问题，它将PCR推广到了更实际的场景中。本文通过利用私有信息检索技术提出了两种I-PCR方案，并分析了其通信开销。此外，我们量化了用户从数据库中获取的信息量，并对所提方案进行了比较。