A Theory of Interpretable Approximations

• URL: http://arxiv.org/abs/2406.10529v1
• Date: Sat, 15 Jun 2024 06:43:45 GMT
• Title: A Theory of Interpretable Approximations
• Authors: Marco Bressan, Nicolò Cesa-Bianchi, Emmanuel Esposito, Yishay Mansour, Shay Moran, Maximilian Thiessen,
• Abstract summary: We study the idea of approximating a target concept $c$ by a small aggregation of concepts from some base class $mathcalH$. For any given pair of $mathcalH$ and $c$, exactly one of these cases holds: (i) $c$ cannot be approximated by $mathcalH$ with arbitrary accuracy. 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-
• Score: 61.90216959710842
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $\mathcal{H}$.

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