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## Model Interpretability through the Lens of Computational Complexity

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Pablo Barceló
• Function : Author
Mikaël Monet
• Function : Author
• PersonId : 1085321
Jorge A. Perez
• Function : Author
• PersonId : 998096

#### Abstract

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 to formally compare among different classes of models. We make a step towards such a notion by studying whether folklore interpretability claims have a correlate in terms of computational complexity theory. We focus on local 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 $\mathcal{C}_1$ of models is more interpretable than another class $\mathcal{C}_2$, if the computational complexity of answering post-hoc queries for models in $\mathcal{C}_2$ is higher than for those in $\mathcal{C}_1$. 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$\neq$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.

### Dates and versions

hal-03052508 , version 1 (10-12-2020)

### Identifiers

• HAL Id : hal-03052508 , version 1
• ARXIV :

### Cite

Pablo Barceló, Mikaël Monet, Jorge A. Perez, Bernardo Subercaseaux. Model Interpretability through the Lens of Computational Complexity. NeurIPS 2020, Dec 2020, Held online, United States. ⟨hal-03052508⟩

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