[论文解读] Faith-Shap: The Faithful Shapley Interaction Index

Che-Ping Tsai, Chih‐Kuan Yeh+1 more|arXiv (Cornell University)|Mar 2, 2022

Bayesian Modeling and Causal Inference被引用 20

一句话总结

本文定义了一种独特的 Faithful Shapley Interaction 指数（Faith-Shap），通过忠实线性近似将 Shapley 值扩展到特征交互，满足自然的交互公理，并提供高效的计算和性质。

ABSTRACT

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.

研究动机与目标

• 动机：在可解释性 AI 中需要对超越单个特征的特征交互进行归因。
• 通过将 Shapley 风格的忠实性扩展到高阶交互，引入 Faith-Interaction 框架。
• 在交互公理下推导出唯一的 Faithful Shapley（Faith-Shap）和 Faithful Banzhaf（Faith-Banzhaf）指数组合。
• 将 Faith-Shap 与现有的交互指数组进行比较，并探讨代数性质及计算效率。

提出的方法

• 将解释建模为一个加权回归，将 v(S) 近似为至多阶 ell 的交互项之和。
• 使用 Möbius 变换 a(v, S) 表示贡献，并推导交互指数的闭式表达。
• 将 Faith-Interaction 指数定义为带有适当权重函数 μ(S) 的加权最小二乘目标的解。
• 证明在合适的 μ(S) 下（以及有限/∞ 约束），Faith-Interaction 指数满足交互线性、对称性与 dummy 公理。
• 通过强制执行交互效率公理专门化为 Faith-Shap，得到一个特定的 μ(S) 和闭式公式（Eq. 16）。
• 证明当 ell = 1 时，Faith-Shap 简化为标准 Shapley 值，并讨论与 Faith-Banzhaf（基于 ell）及其他指数组合的关系。

实验结果

研究问题

• RQ1 如何在保持自然公理的同时，唯一地将类似 Shapley 的归因从单一特征扩展到特征交互？
• RQ2 基于忠实性的替代方法是否能够产生一个唯一的交互指数组（Faith-Shap），满足线性性、对称性、dummy 和效率？
• RQ3 在理论和计算方面，Faithful Shapley 及相关指数组与现有交互指数组的比较？
• RQ4 在估计至选定阶数的交互时，Faith-Shap 的代数性质和实际计算优势是什么？

主要发现

• 通过将交互线性、对称性、dummy 和效率与“自证忠实”的框架相结合，得到唯一的 Faithful Shapley Interaction 指数（Faith-Shap）。
• 当权重函数有限时，Faith-Interaction 类给出闭式解，表现为通过 Möbius 变换和特征事件矩阵 p(S) 表达的形式。
• Faith-Shap 将 Shapley 值推广到交互，在最大交互阶数为 1 时简化为标准 Shapley 值。
• 在广义的 2-效率下，出现互补的 Faith-Banzhaf 指数，具有自身的闭式表达并与 Möbius 变换相关。
• 该框架通过加权线性回归公式实现计算高效的估计，并提供与基数索引和多线性扩展相关的代数表征。