A Theory of Interpretable Approximations
| Autor: | Bressan, Marco, Cesa-Bianchi, Nicolò, Esposito, Emmanuel, Mansour, Yishay, Moran, Shay, Thiessen, Maximilian |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Can a deep neural network be approximated by a small decision tree based on simple features? This question and its variants are behind the growing demand for machine learning models that are *interpretable* by humans. In this work we study such questions by introducing *interpretable approximations*, a notion that captures the idea of approximating a target concept $c$ by a small aggregation of concepts from some base class $\mathcal{H}$. In particular, we consider the approximation of a binary concept $c$ by decision trees based on a simple class $\mathcal{H}$ (e.g., of bounded VC dimension), and use the tree depth as a measure of complexity. Our primary contribution is the following remarkable trichotomy. For any given pair of $\mathcal{H}$ and $c$, exactly one of these cases holds: (i) $c$ cannot be approximated by $\mathcal{H}$ with arbitrary accuracy; (ii) $c$ can be approximated by $\mathcal{H}$ with arbitrary accuracy, but there exists no universal rate that bounds the complexity of the approximations as a function of the accuracy; or (iii) there exists a constant $\kappa$ that depends only on $\mathcal{H}$ and $c$ such that, for *any* data distribution and *any* desired accuracy level, $c$ can be approximated by $\mathcal{H}$ with a complexity not exceeding $\kappa$. This taxonomy stands in stark contrast to the landscape of supervised classification, which offers a complex array of distribution-free and universally learnable scenarios. We show that, in the case of interpretable approximations, even a slightly nontrivial a-priori guarantee on the complexity of approximations implies approximations with constant (distribution-free and accuracy-free) complexity. We extend our trichotomy to classes $\mathcal{H}$ of unbounded VC dimension and give characterizations of interpretability based on the algebra generated by $\mathcal{H}$. Comment: To appear at COLT 2024 |
| Databáze: | arXiv |
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