  To date, there has been no formal study of the statistical cost of
interpretability in machine learning. As such, the discourse around potential
trade-offs is often informal and misconceptions abound. In this work, we aim to
initiate a formal study of these trade-offs. A seemingly insurmountable
roadblock is the lack of any agreed upon definition of interpretability.
Instead, we propose a shift in perspective. Rather than attempt to define
interpretability, we propose to model the \emph{act} of \emph{enforcing}
interpretability. As a starting point, we focus on the setting of empirical
risk minimization for binary classification, and view interpretability as a
constraint placed on learning. That is, we assume we are given a subset of
hypothesis that are deemed to be interpretable, possibly depending on the data
distribution and other aspects of the context. We then model the act of
enforcing interpretability as that of performing empirical risk minimization
over the set of interpretable hypotheses. This model allows us to reason about
the statistical implications of enforcing interpretability, using known results
in statistical learning theory. Focusing on accuracy, we perform a case
analysis, explaining why one may or may not observe a trade-off between
accuracy and interpretability when the restriction to interpretable classifiers
does or does not come at the cost of some excess statistical risk. We close
with some worked examples and some open problems, which we hope will spur
further theoretical development around the tradeoffs involved in
interpretability.
