Consistent Regression using Data-Dependent Coverings

Abstract: In this paper, we introduce a novel method to generate interpretable regression function estimators. The idea is based on called data-dependent coverings. The aim is to extract from the data a covering of the feature space instead of a partition. The estimator predicts the empirical conditional expectation over the cells of the partitions generated from the coverings. Thus, such estimator has the same form as those issued from data-dependent partitioning algorithms. We give sufficient conditions to ensure the consistency, avoiding the sufficient condition of shrinkage of the cells that appears in the former literature. Doing so, we reduce the number of covering elements. We show that such coverings are interpretable and each element of the covering is tagged as significant or insignificant. The proof of the consistency is based on a control of the error of the empirical estimation of conditional expectations which is interesting on its own.