[논문 리뷰] Faith-Shap: The Faithful Shapley Interaction Index

Che-Ping Tsai, Chih‐Kuan Yeh외 1명|arXiv (Cornell University)|2022. 03. 02.

Bayesian Modeling and Causal Inference인용 수 20

한 줄 요약

이 논문은 신뢰할 수 있는 Faithful Shapley Interaction 지수(Faith-Shap)를 정의하여, 자연스러운 상호작용 공리 하에서 신뢰할 수 있는 선형 근사를 통해 Shapley 값을 특징 간 상호작용으로 확장하고, 효율적인 계산과 특성을 제공한다.

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.

연구 동기 및 목표

• 설명 가능한 AI에서 개별 특징을 넘어서는 특징 간 상호작용에 대한 기여도 필요성을 제시한다.
• Shapley 스타일의 충실성을 고차 상호작용으로 확장하여 Faith-Interaction 프레임워크를 도입한다.
• 상호작용 공리하에서 고유한 Faithful Shapley(Faith-Shap)와 Faithful Banzhaf(Faith-Banzhaf) 지수를 도출한다.
• 기존의 상호작용 지수와 Faith-Shap를 비교하고 대수적 특성 및 계산 효율성을 탐구한다.]
• method ["모형 설명을 가중 회귀로 모델링하여 v(S)를 ell 차수까지의 상호작용 항의 합으로 근사한다."
• "Möbius 변환 a(v, S)를 사용하여 기여도를 나타내고 상호작용 지수에 대한 닫힌 형식 표현을 도출한다."
• "적절한 가중 함수 μ(S)로 가중 최소제곱 목적함수의 해로 Faith-Interaction 지수를 정의한다."
• "적절한 μ(S) (유한/무한 제약 하에서)에서 상호작용 선형성, 대칭성 및 더미 공리를 만족함을 입증한다."
• "상호작용 효율성 공리를 적용하여 Faith-Shap로 구체화하고, 특정 μ(S)와 닫힌 형식 식(Eq. 16)을 얻는다."
• "ell = 1일 때 Faith-Shap가 표준 Shapley 값으로 축소되는 것을 보이고, ell 기반의 Faith-Banzhaf 및 다른 지수와의 관계를 논의한다."]
• research_questions:[

제안 방법

• Model explanations as a weighted regression to approximate v(S) by sums of interaction terms up to order ell.
• Use Möbius transform a(v, S) to represent contributions and derive closed-form expressions for interaction indices.
• Define Faith-Interaction indices as solutions to a weighted least squares objective with a proper weighting function μ(S).
• Prove that Faith-Interaction indices satisfy interaction linearity, symmetry, and dummy axioms under suitable μ(S) (and finite/∞ constraints).
• Specialize to Faith-Shap by enforcing the interaction efficiency axiom, yielding a specific μ(S) and a closed-form formula (Eq. 16).
• Show that Faith-Shap reduces to standard Shapley values when ell = 1, and discuss relationships to Faith-Banzhaf (ell-based) and other indices.

실험 결과

연구 질문

• RQ1 How can we uniquely extend Shapley-like attributions from individual features to feature interactions while preserving natural axioms?
• RQ2 Can a faithfulness-based alternative yield a unique interaction index (Faith-Shap) that satisfies linearity, symmetry, dummy, and efficiency?
• RQ3 How do Faithful Shapley and related indices compare to existing interaction indices in theory and computation?
• RQ4 What are the algebraic properties and practical computational benefits of Faith-Shap in estimating interactions up to a chosen order?

주요 결과

• A unique Faithful Shapley Interaction index (Faith-Shap) is obtained by combining interaction linearity, symmetry, dummy, and efficiency with a faithful-by-construction framework.
• The Faith-Interaction class yields closed-form solutions when weighting functions are finite, expressed via Möbius transforms and a feature-incidence matrix (p(S)).
• Faith-Shap generalizes Shapley values to interactions and reduces to standard Shapley values when the maximum interaction order is one.
• A complementary Faith-Banzhaf index arises under generalized 2-efficiency, with its own closed-form and relationships to the Möbius transform.
• The framework enables computationally efficient estimation through a weighted linear regression formulation, and provides algebraic characterizations linked to cardinal indices and multilinear extensions.