  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 re trained. 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.
