基于SHAP分数的简化复杂性研究：通过知识编译的可追溯性和非近似性结果
On the Complexity of SHAP-Score-Based Explanations: Tractability via Knowledge Compilation and Non-Approximability Results
摘要：
在机器学习中，$\mathsf{SHAP}$ -score是Shapley值的一个版本，用于通过为每个特征分配分数来解释特定实体上学习模型的结果。虽然在一般计算Shapley值是一个棘手的问题，我们证明了一个强有力的积极的结果，说明$\mathsf{SHAP}$分数可以计算在多项式时间确定性和可分解的布尔电路。这种电路在知识编译领域中被研究，并且推广了广泛的布尔电路和二元决策图类，包括二元决策树和有序二元决策图（OBDD）。我们还建立了SHAP分数的计算限制，通过观察，计算它在一类布尔模型总是多项式一样难的模型计数问题，该类。这意味着决定性和可分解性都是我们考虑的电路的基本属性。这也意味着，计算$\mathsf{SHAP}$ -分数是棘手的，以及在类的命题公式在DNF。基于这个否定的结果，我们寻找存在的全多项式随机近似计划（FPRAS）计算$\mathsf{SHAP}$ -分数在这样的类。相反，DNF公式的模型计数问题，它承认一个FPRAS，我们证明了没有这样的FPRAS存在的计算$\mathsf{SHAP}$ -分数。令人惊讶的是，这个负结果甚至对DNF中的单调公式类也成立。这些技术可以进一步扩展以证明另一个强否定结果：在广泛相信的复杂性假设下，没有多项式时间算法来检查给定单调DNF公式$φ$和特征$x,y$，$φ$中的$x$的$\mathsf{SHAP}$分数是否小于$φ$中的$y$的$\mathsf{SHAP}$分数。
英文摘要：
In Machine Learning, the $\mathsf{SHAP}$-score is a version of the Shapley value that is used to explain the result of a learned model on a specific entity by assigning a score to every feature. While in general computing Shapley values is an intractable problem, we prove a strong positive result stating that the $\mathsf{SHAP}$-score can be computed in polynomial time over deterministic and decomposable Boolean circuits. Such circuits are studied in the field of Knowledge Compilation and generalize a wide range of Boolean circuits and binary decision diagrams classes, including binary decision trees and Ordered Binary Decision Diagrams (OBDDs). We also establish the computational limits of the SHAP-score by observing that computing it over a class of Boolean models is always polynomially as hard as the model counting problem for that class. This implies that both determinism and decomposability are essential properties for the circuits that we consider. It also implies that computing $\mathsf{SHAP}$-scores is intractable as well over the class of propositional formulas in DNF. Based on this negative result, we look for the existence of fully-polynomial randomized approximation schemes (FPRAS) for computing $\mathsf{SHAP}$-scores over such class. In contrast to the model counting problem for DNF formulas, which admits an FPRAS, we prove that no such FPRAS exists for the computation of $\mathsf{SHAP}$-scores. Surprisingly, this negative result holds even for the class of monotone formulas in DNF. These techniques can be further extended to prove another strong negative result: Under widely believed complexity assumptions, there is no polynomial-time algorithm that checks, given a monotone DNF formula $φ$ and features $x,y$, whether the $\mathsf{SHAP}$-score of $x$ in $φ$ is smaller than the $\mathsf{SHAP}$-score of $y$ in $φ$.
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