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To date, there has been no formal study of the statistical cost ofinterpretability in machine learning. As such, the discourse around potentialtrade-offs is often informal and misconceptions abound. In this work, we aim toinitiate a formal study of these trade-offs. A seemingly insurmountableroadblock is the lack of any agreed upon definition of interpretability.Instead, we propose a shift in perspective. Rather than attempt to defineinterpretability, we propose to model the \emph{act} of \emph{enforcing}interpretability. As a starting point, we focus on the setting of empiricalrisk minimization for binary classification, and view interpretability as aconstraint placed on learning. That is, we assume we are given a subset ofhypothesis that are deemed to be interpretable, possibly depending on the datadistribution and other aspects of the context. We then model the act ofenforcing interpretability as that of performing empirical risk minimizationover the set of interpretable hypotheses. This model allows us to reason aboutthe statistical implications of enforcing interpretability, using known resultsin statistical learning theory. Focusing on accuracy, we perform a caseanalysis, explaining why one may or may not observe a trade-off betweenaccuracy and interpretability when the restriction to interpretable classifiersdoes or does not come at the cost of some excess statistical risk. We closewith some worked examples and some open problems, which we hope will spurfurther theoretical development around the tradeoffs involved ininterpretability.

Enforcing Interpretability and its Statistical Impacts: Trade-offs between Accuracy and Interpretability