Approximation Trees: Statistical Stability in Model Distillation

Approximation Trees: Statistical Stability in Model Distillation Yichen Zhou, Zhengze Zhou, Giles Hooker Department of Statistical Science Cornell University Ithaca, NY 14853, USA Abstract This paper examines the stability of learned explanations for black-box predictions via model distillation with decision trees. One approach to intelligibility in machine learning is to use an understandable "student" model to mimic the output of an accurate "teacher". Here, we consider the use of regression trees as a student model, in which nodes of the tree can be used as "explanations" for particular predictions, and the whole structure of the tree can be used as a global representation of the resulting function. However, individual trees are sensitive to the particular data sets used to train them, and an interpretation of a student model may be suspect if small changes in the training data have a large effect on it. In this context, access to outcomes from a teacher helps to stabilize the greedy splitting strategy by generating a much larger corpus of training examples than was originally available. We develop tests to ensure that enough examples are generated at each split so that the same splitting rule would be chosen with high probability were the tree to be retrained. Further, we develop a stopping rule to indicate how deep the tree should be built based on recent results on the variability of Random Forests when these are used as the teacher. We provide concrete examples of these procedures on the CAD-MDD and COMPAS data sets. 1 Introduction This paper examines the use of regression trees for model distillation. While Machine Learning has traditionally focused on predictive performance, there has been considerable recent interest in "X-raying the black box": finding methods to make the ways in which neural networks, Random Forests and other predictive models arrive at their predictions understandable to humans. This problem can be approached by creating summaries of these models such as variable importance scores (Breiman, 2001), partial dependence or ICE plots (Friedman, 2001; Goldstein et al., 2013), saliency maps (Simonyan et al., 2013) and other local explanations (Ribeiro et al., 2016). It can also be approached by developing intelligible "student" models which mimic the predictions of the original "teacher" black box: a strategy encompassed by the term model distillation . Within model distillation, common student models are generalized additive models (GAMS: see Lou et al. (2012); Tan et al. (2017), Hooker (2007) provides a link between these and PDPs) and decision trees Breiman et al. (1984); Quinlan (1987), which are our focus. Decision trees have an intelligible graphical representation and can automatically fit complex high-dimensional functions, both of which make them appealing as student models.