[论文解读] On Computing Probabilistic Explanations for Decision Trees

Marcelo Arenas, Pablo Barceló+2 more|arXiv (Cornell University)|Jun 30, 2022

Explainable Artificial Intelligence (XAI)被引用 13

一句话总结

本文研究了决策树中计算概率解释——具体而言，δ-充分原因——的计算复杂性，其中必须通过特征子集使模型预测的概率至少达到δ。本文证明了寻找最小大小或包含意义上的最小δ-充分原因的问题是NP难的，解决了两个开放问题，并将布尔电路的困难性结果扩展到更受限的决策树情形，同时识别出可解的结构约束，并提出基于SAT的实用解决方案。

ABSTRACT

Formal XAI (explainable AI) is a growing area that focuses on computing explanations with mathematical guarantees for the decisions made by ML models. Inside formal XAI, one of the most studied cases is that of explaining the choices taken by decision trees, as they are traditionally deemed as one of the most interpretable classes of models. Recent work has focused on studying the computation of "sufficient reasons", a kind of explanation in which given a decision tree $T$ and an instance $x$, one explains the decision $T(x)$ by providing a subset $y$ of the features of $x$ such that for any other instance $z$ compatible with $y$, it holds that $T(z) = T(x)$, intuitively meaning that the features in $y$ are already enough to fully justify the classification of $x$ by $T$. It has been argued, however, that sufficient reasons constitute a restrictive notion of explanation, and thus the community has started to study their probabilistic counterpart, in which one requires that the probability of $T(z) = T(x)$ must be at least some value $δ\in (0, 1]$, where $z$ is a random instance that is compatible with $y$. Our paper settles the computational complexity of $δ$-sufficient-reasons over decision trees, showing that both (1) finding $δ$-sufficient-reasons that are minimal in size, and (2) finding $δ$-sufficient-reasons that are minimal inclusion-wise, do not admit polynomial-time algorithms (unless P=NP). This is in stark contrast with the deterministic case ($δ= 1$) where inclusion-wise minimal sufficient-reasons are easy to compute. By doing this, we answer two open problems originally raised by Izza et al. On the positive side, we identify structural restrictions of decision trees that make the problem tractable, and show how SAT solvers might be able to tackle these problems in practical settings.

研究动机与目标

• 解决决策树中δ-充分原因的计算复杂性问题，这是确定性充分原因的概率松弛形式。
• 解决Izza等人提出的关于最小δ-充分原因可解性问题的两个开放问题。
• 将Wäldchen等人针对布尔电路的困难性结果扩展到更受限的决策树类别。
• 识别出使δ-充分原因计算变得可解的结构约束。
• 探索在现实场景中利用SAT求解器求解δ-充分原因问题的实用方法。

提出的方法

• 将δ-充分原因形式化为特征子集，使得在随机补全下维持相同预测的概率至少为δ。
• 将已知的NP难问题归约到δ-充分原因问题，以证明最小大小和包含意义上的最小变体的NP完全性。
• 引入结构约束（如有界宽度、有界深度）使问题变得可解，并支持多项式时间计算。
• 利用SAT求解器实现满足所识别可解条件的实例的实用解决方案。
• 基于MNIST构建的决策树进行实验评估，可视化并验证所计算的δ-充分原因的大小和结构。
• 分析在真实图像分类任务中，解释大小与置信度阈值δ之间的权衡。

实验结果

研究问题

• RQ1 计算决策树中最小大小的δ-充分原因是否在计算上是可解的，还是NP难的？
• RQ2 即使δ < 1，计算包含意义上的最小δ-充分原因是否也是NP难的？
• RQ3 对决策树施加结构约束是否可使δ-充分原因的计算变得可解？
• RQ4 布尔电路的困难性结果如何扩展到更受限的决策树情形？
• RQ5 在满足可解结构假设的前提下，SAT求解器能否在实践中有效计算δ-充分原因？

主要发现

• 即使δ < 1，计算最小大小的δ-充分原因对决策树而言也是NP难的。
• 计算包含意义上的最小δ-充分原因同样是NP难的，这与确定性情形（δ = 1）不同，在后者中该问题是可解的。
• 即使决策树具有有界深度或有界宽度，该问题仍为NP难，表明其固有的复杂性。
• 结构约束（如宽度或深度有界）可使问题变得可解，从而支持多项式时间计算。
• 基于MNIST构建的决策树的实验结果表明，δ-充分原因的大小可以很小（例如，k* = 5至49个特征），具体取决于数字和模型。
• 当满足结构约束时，SAT求解器在实践中可有效用于计算δ-充分原因。