Shapley Feature Values: Methods & Applications

• Shapley Feature Values (SFV) are attributions derived from cooperative game theory that quantify each feature's contribution to a model's output.
• They employ different coalition value functions—baseline, marginal, and conditional—to provide a range of explanation semantics and estimation strategies.
• Recent advances address computational challenges and feature dependencies, enhancing robust model explanations and practical feature selection.

to=arxiv_search 手机版天天中彩票_search 天天大奖彩票站 պարզ {"query":"all:\"Shapley feature values\" OR ti:\"Shapley value\" explainability feature attribution", "max_results": 10, "sort_by": "relevance"} to=arxiv_search nammineq_search պարզ {"query":"ti:\"Fast Shapley Value Estimation: A Unified Approach\" OR id:(Zhang et al., 2023)", "max_results": 5, "sort_by": "relevance"} Shapley Feature Values (SFV) are feature attributions obtained by applying the Shapley value from cooperative game theory to a predictive model, so that a model output, a loss, or another task-specific quantity is decomposed into feature-wise contributions. In the standard local setting, features are treated as players, a coalition value function $v(S)$ evaluates the model when only a subset $S$ is present, and the attribution of feature $i$ is the weighted average of its marginal contributions over all coalitions. Across the recent literature, SFV has developed into a family of explanation types rather than a single method, because the attribution depends on how missing features are represented, what value function is chosen, and how the exponentially large coalition space is estimated or approximated (Chen et al., 2022, Zhang et al., 2023).

1. Formal game-theoretic definition

For a feature set $N=\{1,\dots,d\}$ and a value function $v:P(N)\to\mathbb{R}$, the exact Shapley value of feature $i$ is

$$\phi_i(v)=\sum_{S\subseteq N\setminus \{i\}} \frac{|S|!(d-|S|-1)!}{d!}\Big(v(S\cup\{i\})-v(S)\Big).$$

This is the standard average marginal-contribution formula used throughout the SFV literature (Zhang et al., 2023). In additive-explanation form, one also writes

$g(\bm{z}') = \phi_0 + \sum_{i=1}^m \phi_i z'_i,$

with local accuracy, missingness, and consistency as the associated SHAP axioms for the simplified binary presence mask $\bm{z}'$ (Michiels et al., 2023, Fujii, 2023).

The standard axiomatic basis of SFV consists of efficiency, symmetry, dummy or null player, and additivity or linearity. In the local explanation setting, efficiency becomes the decomposition

$$f(\boldsymbol{x}^*) = \phi_0 + \sum_{j=1}^M \phi_j,$$

or, in grouped settings, the corresponding sum over groups (Olsen et al., 2021, Jullum et al., 2021). These axioms explain why Shapley values became a widely accepted and trustworthy tool for black-box models, including deep neural networks, tree ensembles, and model-agnostic explainers (Zhang et al., 2023).

A recurrent theme, however, is that the Shapley formula alone does not fix a unique explanation. The same cooperative-game-theoretic definition can generate different SFV depending on the chosen utility $S$0, the masking or replacement rule for absent features, and whether the explanation is intended to be local, global, model-true, or loss-based (Chen et al., 2022, Campbell et al., 2021).

2. Coalition value functions and explanation semantics

A central distinction in SFV is how a coalition $S$1 is mapped to an evaluable model input. The literature distinguishes baseline, marginal, and conditional coalition values. With a fixed baseline $S$2, baseline SFV uses

$S$3

Marginal SFV uses

$S$4

and conditional SFV uses

$S$5

These are not merely computational variants; they correspond to different explanation types (Chen et al., 2022).

This distinction is sharpened by work that separates explanations of the model from explanations of the model combined with feature dependencies in the data. Interventional SHAP is described as explaining the model alone, because it breaks statistical dependence when filling in absent features. Conditional SHAP is described as explaining the result, meaning the model together with the data distribution, because it retains the dependence structure. A direct decomposition of the conditional Shapley contribution into an interventional part and a dependent part is given by

$S$6

so that the direct model effect and the dependence-mediated effect can be examined simultaneously (Michiels et al., 2023).

In decision tree ensembles, the choice of utility is similarly decisive. Observational TreeSHAP and interventional TreeSHAP are presented as two common utilities, while the Eject method defines a different game: follow the decision path for the instance, and if a feature required at a split is not in the coalition, stop early and return the value of the last visited node. This is introduced as a model-true, local explanation that satisfies a stronger local dummy player property: if a feature is not used in prediction for that instance, its attribution should be zero (Campbell et al., 2021).

3. Estimation strategies and computational tractability

The exact computation of SFV is combinatorial. The number of terms in the feature-wise formula is $S$7, and the complexity is exponential in the number of features (Jullum et al., 2021, Zhang et al., 2023). A broad methodological literature therefore separates the problem into two parts: the definition of the coalition game and the tractable estimation strategy (Chen et al., 2022).

A useful synthesis is the unified stochastic-estimator view. Semivalue, random-order value, and least-squares value can all be represented as: sample subsets $S$8, attach coefficients $S$9, and optionally apply a linear transform and bias term. In generic form,

$i$0

Under this view, ApproSemivalue, KernelSHAP, and related methods differ mainly in the subset distribution $i$1, the coefficients $i$2, and the estimator parameterization rather than in the fundamental recipe (Zhang et al., 2023).

The survey of estimation algorithms organizes 24 methods by removal approach and approximation strategy. Equivalent characterizations of the Shapley value motivate semivalue estimators, random-order estimators, weighted least-squares estimators such as KernelSHAP, and multilinear-extension estimators. Model-specific approximations exploit structure in linear models, tree ensembles, or deep networks; interventional TreeSHAP is exact for tree models under the marginal or interventional game, whereas deep-model methods such as DeepLIFT and DeepSHAP are presented as biased approximations (Chen et al., 2022).

Amortized explainers replace per-instance optimization or regression by a learned explainer $i$3. FastSHAP is presented as an amortized least-squares method with a constrained formulation and additive efficient normalization. SimSHAP removes the special metric matrix, the constrained normalization step, and the transformed metric-space fitting, and instead minimizes

$i$4

against an unbiased stochastic target. The reported result is a simpler amortized estimator that is dramatically faster than KernelSHAP and KernelSHAP-S at inference time and slightly faster than FastSHAP while preserving competitive attribution quality (Zhang et al., 2023).

4. Dependence, multicollinearity, and model-faithful SFV

A persistent issue in SFV is feature dependence. One line of work argues that standard Shapley values assume feature independence, so when predictors are correlated, the attribution can be distorted because removing one feature changes the apparent contribution of another correlated feature. The proposed multicollinearity-corrected Shapley value constructs an adjustment factor $i$5 so that the corrected feature is uncorrelated with the feature or coalition of interest. For a single feature $i$6,

$i$7

The corrected value is then computed by Monte Carlo on adjusted coalitions, with the claim that the correction factor has almost no effect on execution time relative to the uncorrected approximation (Basu et al., 2020).

A different dependence-aware strategy is to estimate the conditional distributions needed by conditional SFV directly. The VAEAC-based approach uses a variational autoencoder with arbitrary conditioning to model all feature dependencies simultaneously and then approximates coalition values by Monte Carlo samples from the learned conditional distribution. The method is presented as handling continuous, categorical, and mixed data in one framework, with a non-uniform masking scheme that focuses on the coalitions actually queried during Shapley estimation and substantially improves performance in high-dimensional settings (Olsen et al., 2021).

The literature also emphasizes that dependence and local model faithfulness are not interchangeable. In tree ensembles, Eject is introduced specifically because TreeSHAP-style utilities can assign non-zero attribution to features not used in the model’s actual decision path. The resulting contrast is between data-true explanations, which reflect broader information patterns in the data, and model-true explanations, which reflect the specific prediction algorithm for the individual instance (Campbell et al., 2021).

5. Grouped, joint, and set-valued generalizations

Several extensions of SFV replace the singleton feature as the basic explanatory unit. groupShapley computes Shapley values directly on a partition of the feature set into non-overlapping, non-empty groups. It is defined as a proper Shapley value computation on a game whose players are feature groups, not as a heuristic aggregation of feature-wise values. Under partial additive separability,

$i$8

and group independence, post-grouped Shapley equals groupShapley. Outside those conditions, the two quantities may differ. The computational motivation is immediate: the complexity drops from $i$9 to $N=\{1,\dots,d\}$0, a relative reduction of $N=\{1,\dots,d\}$1; for $N=\{1,\dots,d\}$2 and $N=\{1,\dots,d\}$3, the paper gives a reduction $N=\{1,\dots,d\}$4 (Jullum et al., 2021).

Grouped Shapley Values for tree ensembles are defined similarly as direct group-level Shapley values over predefined coalitions $N=\{1,\dots,d\}$5,

$N=\{1,\dots,d\}$6

The literature stresses that naive summation of individual Shapley values within a group is not theoretically sound in general. For grouped tree explanations, an efficient algorithm with runtime

$N=\{1,\dots,d\}$7

is given for random forests and XGBoost-style models (Huber et al., 2023).

Joint Shapley values extend the attribution target from single features to coalitions $N=\{1,\dots,d\}$8. The defining formula is

$N=\{1,\dots,d\}$9

with a uniqueness theorem for every explanation order $v:P(N)\to\mathbb{R}$0. The stated interpretation is that joint Shapley measures a set’s own average contribution, whereas interaction indices measure how a set behaves relative to its members (Harris et al., 2021).

Shapley Sets changes the player set itself by first decomposing the feature space into non-separable variable groups using a recursive decomposition algorithm with runtime $v:P(N)\to\mathbb{R}$1. Each group is then treated as a super-feature, and the attribution equals the Shapley value over the transformed feature set. This is presented as advantageous when conventional SFV is misleading due to feature interaction in both the model and the data (Sivill et al., 2023).

6. Feature selection and task-specific SFV

The use of SFV for feature selection is both productive and contested. A direct critique is that the Shapley axioms are about fair allocation of payoff in a cooperative game, not about selecting an optimal subset of features. On this view, a high Shapley score can belong to a feature that is not part of the best model, and a crucial feature for the best model can receive a lower Shapley score because the Shapley value averages marginal contributions over all coalitions rather than contributions to the best submodel (Fryer et al., 2021).

Despite that critique, several task-specific SFV-based feature selection methods have been proposed. LLpowershap is a wrapper-style feature selection method for binary classification that uses loss-based Shapley values rather than prediction-based Shapley values. It defines the cooperative game on logistic loss, computes LogisticLossSHAP on a test split, compares each real feature to the strongest of five synthetic noise features, and selects features via the Mann–Whitney U test with threshold $v:P(N)\to\mathbb{R}$2. It is explicitly positioned as a loss-based Shapley Feature Value wrapper for automated feature selection (Madakkatel et al., 2024).

For regression under concept shift, SHAPEffects uses local Shapley values together with validation-set prediction errors. Observations are divided into correctly predicted, under predicted, and over predicted groups, and a feature is removed if its local Shapley contributions systematically push predictions in the wrong direction relative to the observed error. The method is presented as robust in both shifted and static settings, with the strongest gains in synthetic drift tests and in electricity price forecasting and housing-market cases (Sebastián et al., 2023).

A different formulation is the classification game with hinge-loss-based characteristic function and Shapley-value-based error apportioning (SVEA). Here the game value is the reduction in hinge training error achieved by a coalition of features, and the sign of the resulting allocation is used for selection: thresholding SVEA at 0 identifies a subset with strong joint contribution to classification and, in some examples, the effective low-dimensional subspace where the data predominantly lies (Tripathi et al., 2020).

SFV has also been used as an offline feature-selection mechanism in tactile-internet signal prediction. In that setting, SFV ranks historical haptic features and is combined with a Gaussian Process oracle and a neural predictor. The reported best result is GP+SFV with ResNet at 95.72% accuracy and 2.2 ms average inference time, with additional claims of large inference-time reductions relative to LeFo and GP baselines (Vahedifar et al., 25 Sep 2025).

7. Critiques, privacy risks, and unresolved questions

Recent work has made explicit that SFV is not only an interpretability tool but also a potential leakage channel. In the feature-inference attack setting, a model service returns both the prediction and a Shapley explanation vector for a private query, and the attacker attempts to reconstruct the input from the explanation. On Google Cloud, Microsoft Azure, IBM aix360, and a vanilla implementation, the attacks are reported to work with surprisingly few queries: with about 100 queries, attack 1 reconstructs private features with about 10% average deviation from ground truth; under the no-background-knowledge setting, attack 2 reconstructs at least 30% of features with about 14% average deviation (Luo et al., 2024).

This privacy result reinforces a more general conceptual point: SFV is not unique until the game is specified. For tree ensembles, the literature explicitly argues that “the Shapley values that can be generated for even a single machine learning model are far from unique,” because theoretical and implementational decisions affect the resulting attributions. Different utilities answer different questions, so disagreement between methods is not necessarily numerical error; it can reflect a difference in the object being explained (Campbell et al., 2021).

Open methodological gaps remain. Conditional SFV is still difficult because the conditional expectation

$v:P(N)\to\mathbb{R}$3

is intrinsically hard to estimate in high dimensions. Deep-model estimators remain largely approximate, and the literature calls for better convergence diagnostics, stronger surrogate and generative models for conditional estimation, and clearer reporting of explanation type, baseline distribution, estimation strategy, and convergence diagnostics (Chen et al., 2022). A plausible implication is that SFV research is increasingly organized around three linked questions: what coalition game should be explained, how that game can be estimated tractably, and what trade-off between faithfulness, usability, and privacy is acceptable in a given application.