Faithful Shapley Interaction Index

• Faith–Shap is a mathematically principled framework that generalizes the Shapley value to include interaction effects while preserving efficiency, linearity, symmetry, and dummy properties.
• It employs a weighted least-squares approximation over set functions and leverages the Möbius transform to uniquely determine interaction attributions for feature subsets.
• The method offers computational advantages in practical machine learning applications, supporting efficient estimators and scalable approximations for model interpretability and sensitivity analysis.

The Faithful Shapley Interaction Index (often abbreviated as Faith–Shap) is a mathematically principled generalization of the classical Shapley value to feature interactions in cooperative games and explainable machine learning. Rooted in the axiomatic foundations of the original Shapley value, Faith–Shap extends attribution beyond main effects to interaction terms, ensuring that the efficiency, linearity, symmetry, and dummy properties are preserved for arbitrary orders of interaction. Faith–Shap emerges as the unique solution to a weighted least-squares approximation problem over set functions, providing both theoretical rigor and computational benefits in decomposing complex predictive models into interpretable contributions from feature subsets.

1. Theoretical Foundation and Motivation

The Faithful Shapley Interaction Index (Faith–Shap) aims to allocate attributions not only to individual features but also to higher-order interactions between feature groups in a way that remains faithful to the underlying model's behavior and adheres to natural extensions of the canonical Shapley axioms. Traditional Shapley values, designed for single-player attributions, are uniquely characterized by linearity, symmetry, dummy, and efficiency properties over individual features. However, direct extension of these axioms to subsets of features does not yield a unique interaction index, prompting the need for a new principled formulation (Tsai et al., 2022).

Faith–Shap addresses this by posing the attribution problem as one of finding the best (in a weighted least-squares sense) approximation of the model's set function by an $\ell$-order polynomial, where interaction coefficients act as generalized Shapley values for sets of cardinality up to $\ell$. This "faithfulness" requirement, alongside axiomatically extended linearity, symmetry, dummy, and efficiency, singles out Faith–Shap as the canonical interaction index.

2. Axiomatic Characterization

Faith–Shap is founded on the interaction-extended versions of the four central Shapley value axioms:

• Interaction Linearity: The index is linear in the set function; if $v = \alpha v_1 + \beta v_2$, then Faith–Shap is the corresponding linear combination of the interaction indices for $v_1$ and $v_2$.
• Interaction Symmetry: If two features or two groups are interchangeable within the model, their attributions must be equal.
• Interaction Dummy: If a feature does not affect the model output in any coalition, all interactions containing it are assigned zero attribution.
• Interaction Efficiency: The sum of all interaction attributions up to order $\ell$ matches the total value assigned to the full set, i.e., $v([d]) - v(\emptyset)$ (Tsai et al., 2022).

These axioms are satisfied exclusively by Faith–Shap in the context of $\ell$-order polynomial approximations, in contrast to alternative indices (e.g., Shapley-Taylor, SII) which may require additional distribution axioms or sacrifice efficiency (Dhamdhere et al., 2019).

3. Mathematical Formulation

Let $v$ be a pseudo-Boolean function (set function on the power set of $[d]$). The Faith–Shap index seeks coefficients $_S(v, \ell)$ for subsets $S$ of size at most $\ell$ such that:

$$\min \sum_{A \subseteq [d]} \mu(A) \left[v(A) - \sum_{T \subseteq A, |T| \leq \ell} {}_T(v, \ell)\right]^2$$

subject to the constraints $v(\emptyset) = {}_\emptyset(v, \ell)$ and $\sum_{T: |T| \leq \ell} {}_T(v, \ell) = v([d])$.

Closed-form solutions are provided in terms of the Möbius transform $$a(v, S) = \sum_{T \subseteq S} (-1)^{|S| - |T|} v(T)$$:

$$\text{F-Shap}_S(v, \ell) = a(v, S) + (-1)^{\ell - |S|} \frac{|S|}{\ell+|S|} \binom{\ell}{|S|} \sum_{T \supset S, |T| > \ell} \frac{\binom{|T|-1}{\ell}}{\binom{|T|+\ell-1}{\ell+|S|}} a(v, T)$$

for all $S$ with $|S| \leq \ell$ (Tsai et al., 2022).

For the highest-order interactions ($|S| = \ell$), the index can be rewritten as a weighted sum of discrete derivatives,

$$\text{F-Shap}_S(v, \ell) = \gamma \sum_{T \subseteq [d] \setminus S} c(T) \cdot \Delta_S(v(T)),$$

with explicit factorial-based coefficients. For $\ell{=}1$, the formula reduces to the classical Shapley value.

4. Comparison with Alternative Interaction Indices

Faith–Shap differs from the Shapley Interaction Index (SII) and Shapley-Taylor Index (STI) fundamentally in its axiomatic basis and the faithfulness of its allocation:

• SII and related indices rely on recursive distribution or interaction distribution axioms and may not satisfy efficiency, causing main and interaction attributions to sum to less than the total model value (Dhamdhere et al., 2019, Fumagalli et al., 2023).
• STI introduces an interaction distribution axiom that avoids "leakage" of pure interaction effects but may over-concentrate attributions at the highest order.
• Faith–Shap uniquely uses only the interaction-extended basic Shapley axioms, avoiding ad hoc rules while guaranteeing additive decompositions up to the desired order and exact recovery for functions decomposable as generalized additive models (GAMs) of order $\ell$ (Bordt et al., 2022).

5. Computational Properties

Although an exact computation of Faith–Shap generally requires exponential time in the number of features, the weighted least-squares structure admits substantial computational benefits in practical settings:

• If the underlying model is sparse (e.g., only low-order interactions are present), polynomial-time computation is possible. For example, functional ANOVA decompositions and $k$-additive games allow for a restriction to interactions of order up to $k$, reducing the number of parameters and coalition evaluations (Gevaert et al., 2022, Hu et al., 2023).
• Efficient estimators and approximation methods, such as surrogate PDD-SHAP models or sampling-based approaches like SHAP-IQ, enable scalable calculation of Faith–Shap indices and provide provable convergence guarantees (Gevaert et al., 2022, Fumagalli et al., 2023).
• Faith–Shap supports additive decomposability: if the model decomposes into independent components over feature groups, the Faith–Shap attribution for each group can be computed and summed independently (Hu et al., 2023).

6. Practical Applications and Interpretability

Faith–Shap provides a principled attribution for both main and interaction effects in a variety of domains:

• In sensitivity analysis, it enables fair allocation of both direct and interaction-driven variance even under correlated inputs, outperforming Sobol' indices in cases of dependence (Iooss et al., 2017, Plischke et al., 2020).
• In model explanation, Faith–Shap supports granular attribution of predictions to subsets of features, allowing disentanglement of individual and combinatorial effects (e.g., medical risk assessment, LLM interpretation) (Nohara et al., 2022, Tsai et al., 2022).
• In software toolchains, Faith–Shap-based explanation frameworks are available for both local and global post-hoc model inspection (e.g., the nshap Python package (Bordt et al., 2022)).

These capabilities empower the detection and interpretation of synergistic or antagonistic feature interactions, providing actionable and domain-relevant insights for model auditing, fairness, and transparency.

7. Limitations, Open Problems, and Directions

Several open challenges and limitations are noted in the literature:

• The computation of Faith–Shap remains exponential in the general case; efficient algorithms rely on model structure, sparsity, or controlled approximation.
• No simple closed-form discrete-derivative representation for lower-order interactions in Faith–Shap is currently known, in contrast with SII/STI (Fumagalli et al., 2023).
• Estimation can become statistically demanding when higher-order interactions are present or in the presence of substantial feature dependence (Bordt et al., 2022).
• Theoretical recovery is exact for models that are generalized additive up to the chosen order; for models with significant higher-order or non-additive structure, interpretations may be less clear.
• Open problems include the extension and validation of Faith–Shap under strong feature dependence, its robustness to sampling noise in high dimensions, and the reconciliation of Faith–Shap with domain-specific notions of explanation (e.g., contrastive or counterfactual attributions).

Faith–Shap represents a precise and functionally grounded framework for decomposing model influences, reconciling efficiency, interpretability, and fairness in feature attribution. It is supported by algorithms, closed-form solutions for special cases, and practical implementations, making it a central tool in modern explainable machine learning.