Model Interpretability through the Lens of Computational Complexity

In spite of several claims stating that some models are more interpretable than others --e.g., "linear models are more interpretable than deep neural networks"-- we still lack a principled notion of interpretability that allows us to formally compare among different classes of models. We make a step towards such a theory by studying whether folklore interpretability claims have a correlate in terms of computational complexity theory. We focus on post-hoc explainability queries that, intuitively, attempt to answer why individual inputs are classified in a certain way by a given model. In a nutshell, we say that a class C1 of models is more interpretable than another class C2, if the computational complexity of answering post-hoc queries for models in C2 is higher than for C1. We prove that this notion provides a good theoretical counterpart to current beliefs on the interpretability of models; in particular, we show that under our definition and assuming standard complexity-theoretical assumptions (such as P!=NP), both linear and tree-based models are strictly more interpretable than neural networks. Our complexity analysis, however, does not provide a clear-cut difference between linear and tree-based models, as we obtain different results depending on the particular {post-hoc explanations} considered. Finally, by applying a finer complexity analysis based on parameterized complexity, we are able to prove a theoretical result suggesting that shallow neural networks are more interpretable than deeper ones.