SHAP Value Attribution

• SHAP value attribution is a game-theoretic framework that uses Shapley values to fairly distribute the prediction difference among input features based on their marginal contributions.
• It employs both model-agnostic and model-specific algorithms, such as KernelSHAP, TreeSHAP, and DeepSHAP, to efficiently compute feature attributions despite combinatorial complexity.
• Widely applied in bias detection, feature selection, and model debugging, SHAP enhances transparency and trust in machine learning systems by adhering to key axioms like efficiency, symmetry, dummy, and additivity.

SHAP value attribution is a game-theoretic framework for quantifying the contribution of individual input features to the output of a predictive model. Originating from the Shapley value in cooperative game theory, SHAP (SHapley Additive exPlanations) attributes the prediction difference between a reference state and the actual input to features, ensuring fair allocation grounded in rigorous axiomatic foundations. SHAP has unified and formalized a broad class of additive feature-importance methods, offering unique guarantees under its axioms, and is now central to interpretable machine learning across diverse domains.

1. Theoretical Foundation: Cooperative Game Theory and the Shapley Value

The Shapley value, introduced by L. S. Shapley, allocates the total value generated by a coalition of players to individual players according to their average marginal contributions across all permutations of coalition formation. Formally, for a model $f : \Omega \rightarrow \mathbb{R}$, a feature set $N = \{1, ..., n\}$, and an instance $x \in \Omega$, SHAP assigns to feature $i$ the value

$$\phi_i(f, x) = \sum_{S\subseteq N\setminus \{i\}} \frac{|S|! \, (n - |S| - 1)!}{n!} \; \left[ v(S \cup \{i\}) - v(S) \right],$$

where $v(S)$ is a suitable value function for the coalition $S$, such as the expected model output when features in $S$ are known and others marginalized.

SHAP adopts the Shapley value as a feature attribution because it uniquely satisfies four axioms proposed for feature importance:

• Efficiency (Completeness): $\sum_{i=1}^n \phi_i(f, x) = v(N) - v(\emptyset)$
• Symmetry (Fairness): Indistinguishable features in $v$ receive equal attributions.
• Dummy (Missingness): Features with no effect get zero attribution.
• Additivity (Linearity): Attributions are linear in the value function.

Extensions for specific modalities—ranking (Chowdhury et al., 2024), time series (Serramazza et al., 3 Sep 2025, Kim et al., 3 Jan 2026), kernel machines (Chau et al., 2021), instance attribution (Wang et al., 2024), and LLM stochastic inference (Naudot et al., 3 Nov 2025)—adapt the value function or axiomatic constraints to encode the statistical or structural properties of the target problem.

2. Algorithms and Computation: Exact and Approximate SHAP

Direct computation of SHAP values involves summing over all $2^{n-1}$ coalitions for each feature. This is intractable for $n \gtrsim 20$. Several algorithmic strategies have been developed (Lundberg et al., 2017, Barceló et al., 4 Jan 2025):

• Model-agnostic methods:
  • KernelSHAP: Samples random coalitions and solves a (weighted) least-squares regression to approximate the Shapley solution (Lundberg et al., 2017). The weighting kernel enforces the desired axioms by matching sampling weights to permutation probabilities.
  • Monte Carlo Permutation Sampling: Draws random permutations, calculates the marginal contribution of each feature as it enters the ordering, and averages over permutations (Moehle et al., 2021, Horovicz, 14 Dec 2025).
  • SIM-Shapley: Stochastic iterative momentum methods improve the variance and stability of the estimator with provable convergence (Fan et al., 13 May 2025).
• Model-specific methods:
  • TreeSHAP: For tree models, leverages tree structure for polynomial-time exact Shapley computation (Lundberg et al., 2017).
  • DeepSHAP: Composes fast, layer-wise approximations using back-propagation rules for neural architectures (Chen et al., 2021).
• Fourier and Spectral approaches: Construct sparse or low-degree surrogate functions in orthonormal bases, compute Shapley values as linear functionals in the spectral domain for dramatic efficiency gains (Gorji et al., 2024, Morales, 31 Oct 2025).

A “simple index” condition on the power index ensures that if a model family allows polynomial-time expectation computation, then SHAP is also polynomial in $n$ (Barceló et al., 4 Jan 2025).

3. Variants, Extensions, and Generalization

Recognizing limitations in the uniform weighting of marginal contributions, WeightedSHAP generalizes SHAP to allow learned (data-driven) weights for coalitions of different sizes, yielding a family of "semivalues" that relax the efficiency axiom for improved fidelity in certain predictive tasks (Kwon et al., 2022).

In structured data (e.g., time series or multi-modal), players can be redefined as higher-level units—segments (Serramazza et al., 3 Sep 2025), group-segments (Kim et al., 3 Jan 2026), tools (Horovicz, 14 Dec 2025), or nodes (Fan et al., 2 Oct 2025). For instance, GroupSegment-SHAP utilizes data-driven clustering (e.g., HSIC for variable grouping) and change-point segmentation (via MMD) to define multivariate-temporal SHAP units, improving faithfulness and interpretability in complex time series (Kim et al., 3 Jan 2026).

Moreover, recent work addresses the challenge of feature dependencies and causality. Causal-SHAP integrates causal discovery (PC algorithm) and intervention effect estimation (IDA), constructing a new value function that restricts marginalization to realistic, DAG-constrained distributions. This discounts attribution to spurious, merely correlated features and improves robustness in scientific domains (Ng et al., 31 Aug 2025).

In LLMs, stochastic inference means deterministic Shapley axioms can be violated unless careful caching is used. The principle satisfaction trade-off for different SHAP approximations in stochastic settings has been carefully enumerated (Naudot et al., 3 Nov 2025).

4. Practical Applications, Limitations, and Robustness

SHAP-based attribution is widely used for:

• Auditing and bias detection: Identifying over-reliance on spurious or protected features, diagnosing OOD failures, or enforcing fairness (e.g., via RKHS-SHAP regularization or Causal SHAP) (Chau et al., 2021, Ng et al., 31 Aug 2025, Yuan et al., 2024).
• Feature selection and model debugging: Quantifying feature redundancy or safely removing globally-unimportant features, provided aggregate SHAP values are computed on the extended product distribution (not the empirical support) (Bhattacharjee et al., 29 Mar 2025).
• Instance-level decision diagnostics: For instance selection, data pruning, and error analysis in large foundation models via robust Shapley instance scores (Wang et al., 2024).
• Explaining models with non-traditional units: Tools in LLM agents (AgentSHAP (Horovicz, 14 Dec 2025)), ranking features in IR systems (RankSHAP (Chowdhury et al., 2024)), time segments (Serramazza et al., 3 Sep 2025, Kim et al., 3 Jan 2026), or neural units in symbolic networks (Fan et al., 2 Oct 2025).
• Distributed/heterogeneous systems: DeepSHAP enables secure attribution across compositions of models with institutional boundaries (Chen et al., 2021).

Limitations include high computational cost in high dimensions (though mitigated through stochastic, Fourier, and surrogate approaches (Gorji et al., 2024, Morales, 31 Oct 2025, Fan et al., 13 May 2025)), vulnerability to adversarial output permutations (Yuan et al., 2024), instability under model or data distribution drift, and possible misattribution in highly-correlated or causally-confounded settings unless specifically addressed (Ng et al., 31 Aug 2025).

An additional concern is the dependence of SHAP values on the reference (or marginalization) distribution; distributional uncertainty can lead to substantial variability in attributions, and computing robust SHAP intervals is generally NP-hard (Cifuentes et al., 2024).

5. Empirical Evaluation and Comparative Performance

Experimental studies have demonstrated that SHAP-based methods often outperform alternatives such as LIME, EXS, or leave-one-out heuristics in correlating with ground-truth importance, accurately recovering model-internal orderings, and producing explanations aligned with human intuition. For example:

• RankSHAP improves Fidelity (Kendall’s τ) by ∼30.8% over the best non-Shapley baselines in ranking tasks; in user studies, it produces more interpretable explanations than LIME-style or EXS baselines (Chowdhury et al., 2024).
• In time series, segment-level SHAP values computed on equal-length segmentations plus per-segment normalization systematically improve explanation quality, outperforming custom segmentation heuristics (Serramazza et al., 3 Sep 2025).
• FreeShap achieves lower sign-robustness error and higher fidelity in data-centric tasks, outperforming influence functions and gradient-based alternatives in LLM scenarios (Wang et al., 2024).
• Spectral SHAP methods deliver 10–10,000× speedup over classical KernelSHAP approaches, often with negligible loss in attribution accuracy for practical surrogate model sizes (Gorji et al., 2024, Morales, 31 Oct 2025).

The comparison table below summarizes major SHAP algorithm variants:

Method	Exactness	Model Scope	Complexity	Key Reference
KernelSHAP	Approx.	Agnostic	O(K·n² + n³)	(Lundberg et al., 2017)
TreeSHAP	Exact	Trees/GBTs	O(T·L·d²)	(Lundberg et al., 2017)
DeepSHAP	Approx.	Deep nets	O(#ops)	(Chen et al., 2021)
FourierSHAP	Approx./Ex	Discrete	O(k·n·m)	(Gorji et al., 2024)
SIM-Shapley	Approx.	Agnostic	O(iter·n²)	(Fan et al., 13 May 2025)
RKHS-SHAP	Exact	Kernel models	O(m·n·d)	(Chau et al., 2021)
WeightedSHAP	Approx.	Agnostic	O(MLE+SHAP)	(Kwon et al., 2022)
RankSHAP	Exact/MC	Rankers	O(K·n log n)	(Chowdhury et al., 2024)
GroupSegmentSHAP	Approx.	MultivariateTS	O(M·cost_fwd)	(Kim et al., 3 Jan 2026)

Empirical evidence indicates that model-specific, data-driven adaptations (RKHS-SHAP, WeightedSHAP, RankSHAP, Causal-SHAP) improve attribution fidelity, interpretability, or statistical robustness for their respective application contexts.

6. Open Directions and Challenges

• Computational scalability: Further algorithmic advances in variance reduction, structure exploitation (Fourier, RKHS), and amortized or streaming settings are active areas of research.
• Distributional robustness: Development of tractable, robust SHAP intervals under uncertainty, including theoretical complexity characterization (Cifuentes et al., 2024).
• Causality and feature dependence: Integration of causal discovery and conditional sampling is not fully resolved, especially for non-tabular or high-dimensional inputs (Ng et al., 31 Aug 2025).
• Defenses against strategic manipulations: Output-shuffling attacks can cause SHAP values to lose fairness-detection power unless augmentations or cross-method checks are performed (Yuan et al., 2024).
• Stochastic inference: Adapting the axiomatic framework to stochastic models, such as LLMs, where full satisfaction of Shapley axioms may require model-level determinization or modified aggregation schemes (Naudot et al., 3 Nov 2025).
• Generalization to non-discrete spaces: Extending spectral approaches and closed-form solutions to continuous domains and hybrid settings remains an open problem (Morales, 31 Oct 2025).

7. Summary and Significance

SHAP value attribution has established itself as a unifying, axiomatically-justified method for feature importance assignment across machine learning. Its theoretical guarantees—uniqueness, fairness, and local accuracy—are matched by an expanding library of efficient algorithms and practical variants tailored to modern data modalities and model classes. SHAP remains an essential tool for transparency, auditability, and principled interpretability in complex predictive systems, continuing to evolve in response to new challenges in high-dimensionality, distribution shift, causality, data-centric AI, and robust machine learning (Lundberg et al., 2017, Chowdhury et al., 2024, Barceló et al., 4 Jan 2025, Serramazza et al., 3 Sep 2025, Wang et al., 2024, Chau et al., 2021, Kim et al., 3 Jan 2026, Morales, 31 Oct 2025, Fan et al., 13 May 2025, Ng et al., 31 Aug 2025).