Interpretable regression trees using conformal prediction

Abstract

A key property of conformal predictors is that they are valid, i.e., their error rate on novel data is bounded by a preset level of confidence. For regression, this is achieved by turning the point predictions of the underlying model into prediction intervals. Thus, the most important performance metric for evaluating conformal regressors is not the error rate, but the size of the prediction intervals, where models generating smaller (more informative) intervals are said to be more efficient. State-of-the-art conformal regressors typically utilize two separate predictive models: the underlying model providing the center point of each prediction interval, and a normalization model used to scale each prediction interval according to the estimated level of difficulty for each test instance. When using a regression tree as the underlying model, this approach may cause test instances falling into a specific leaf to receive different prediction intervals. This clearly deteriorates the interpretability of a conformal regression tree compared to a standard regression tree, since the path from the root to a leaf can no longer be translated into a rule explaining all predictions in that leaf. In fact, the model cannot even be interpreted on its own, i.e., without reference to the corresponding normalization model. Current practice effectively presents two options for constructing conformal regression trees: to employ a (global) normalization model, and thereby sacrifice interpretability; or to avoid normalization, and thereby sacrifice both efficiency and individualized predictions. In this paper, two additional approaches are considered, both employing local normalization: the first approach estimates the difficulty by the standard deviation of the target values in each leaf, while the second approach employs Mondrian conformal prediction, which results in regression trees where each rule (path from root node to leaf node) is independently valid. An empirical evaluation shows that the first approach is as efficient as current state-of-the-art approaches, thus eliminating the efficiency vs. interpretability trade-off present in existing methods. Moreover, it is shown that if a validity guarantee is required for each single rule, as provided by the Mondrian approach, a penalty with respect to efficiency has to be paid, but it is only substantial at very high confidence levels.