Shapley values, which were originally designed to assign attributions to individual players in coalition games, have become a commonly used approach in explainable machine learning to provide attributions to input features for black-box machine learning models. A key attraction of Shapley values is that they uniquely satisfy a very natural set of axiomatic properties. However, extending the Shapley value to assigning attributions to interactions rather than individual players, an interaction index, is non-trivial: as the natural set of axioms for the original Shapley values, extended to the context of interactions, no longer specify a unique interaction index. Many proposals thus introduce additional less ''natural'' axioms, while sacrificing the key axiom of efficiency, in order to obtain unique interaction indices. In this work, rather than introduce additional conflicting axioms, we adopt the viewpoint of Shapley values as coefficients of the most faithful linear approximation to the pseudo-Boolean coalition game value function. By extending linear to $\ell$-order polynomial approximations, we can then define the general family of faithful interaction indices. We show that by additionally requiring the faithful interaction indices to satisfy interaction-extensions of the standard individual Shapley axioms (dummy, symmetry, linearity, and efficiency), we obtain a unique Faithful Shapley Interaction index, which we denote Faith-Shap, as a natural generalization of the Shapley value to interactions. We then provide some illustrative contrasts of Faith-Shap with previously proposed interaction indices, and further investigate some of its interesting algebraic properties. We further show the computational efficiency of computing Faith-Shap, together with some additional qualitative insights, via some illustrative experiments.

翻译：Shapley值最初设计用于为联盟博弈中的个体参与者分配归因，现已广泛应用于可解释机器学习领域，用于为黑箱机器学习模型的输入特征提供归因。Shapley值的一大吸引力在于其唯一满足一组非常自然的公理化性质。然而，将Shapley值扩展为对交互作用（而非个体参与者）分配归因的交互作用指数并非易事：原始Shapley值的自然公理集在交互场景中不再唯一确定一个交互作用指数。因此，许多研究引入了额外的、较不“自然”的公理，同时牺牲了关键的效率公理，以获得唯一的交互作用指数。在本工作中，我们不引入额外的冲突公理，而是采用将Shapley值视为对伪布尔博弈值函数最忠实线性逼近系数的观点。通过将线性逼近扩展至$\ell$阶多项式逼近，我们得以定义广义的忠实交互作用指数族。研究表明，额外要求忠实交互作用指数满足标准个体Shapley公理（哑元性、对称性、线性及效率）的交互扩展后，可得到唯一的忠实Shapley交互作用指数，记作Faith-Shap，它是Shapley值在交互场景下的自然推广。我们通过对比Faith-Shap与既有交互作用指数阐明了其特性，并进一步探讨了其有趣的代数性质。最后通过示例实验，展示了Faith-Shap的计算效率及其带来的定性洞见。