shapley value computation

Kernel methods suffer from poor interpretability, and exact Shapley value computation is typically intractable due to exponential time complexity. Method: This paper proposes PKeX-Shapley, the first algorithm enabling exact polynomial-time Shapley value computation under product kernel models. Its core innovation lies in exploiting the multiplicative structure of product kernels to derive a decomposable functional representation and a recursive Shapley value formula, integrating RKHS theory, functional space decomposition, and dynamic programming for efficiency. Contribution/Results: PKeX-Shapley reduces Shapley value computation complexity from exponential to polynomial time, achieving zero-approximation-error attribution in kernel regression and classification. Moreover, it generalizes to statistical discrepancy measures—including MMD and HSIC—enabling rigorous feature-level interpretability analysis while preserving theoretical fidelity to the underlying kernel model.

Exact Shapley value computation requires exponentially many model evaluations, while existing stochastic estimation methods—such as KernelSHAP—lack non-asymptotic theoretical guarantees. This paper proposes a unified stochastic estimation framework for Shapley values. First, it establishes the first rigorous non-asymptotic error bound for KernelSHAP. Second, it unifies both with- and without-replacement sampling strategies within a single probabilistic model, explicitly characterizing the bias–variance trade-off across estimators. Third, integrating tools from probability sampling theory, combinatorial inference, and non-asymptotic concentration inequalities, it designs a decision-tree-specific implementation with sparse computational optimizations. Experiments demonstrate that our method achieves significantly lower mean squared error than baseline estimators on medium- and small-scale datasets, requiring fewer samples. On MNIST and CIFAR-10, it substantially outperforms the open-source KernelSHAP library in attribution accuracy and efficiency.

Exact computation of Shapley values is generally infeasible in machine learning, and existing efficient approximation methods often lack theoretical guarantees—particularly, the mechanism underlying the effectiveness of paired sampling remains unclear. This work proposes OddSHAP, which for the first time reveals that paired sampling achieves accurate estimation by orthogonalizing the even component while preserving the odd component of the set function. Building on the key insight that Shapley values depend solely on the odd part of the set function, OddSHAP constructs a consistent estimator operating exclusively within the odd subspace. The method leverages Fourier bases to isolate this subspace and integrates a surrogate model to identify critical high-order interactions, thereby circumventing combinatorial explosion. Empirical evaluations across multiple benchmarks demonstrate that OddSHAP achieves state-of-the-art estimation accuracy.

To address the high variance and low computational efficiency of Shapley value approximations in KernelSHAP, this paper proposes a deterministic kernel weighting scheme that replaces conventional stochastic weighting. Grounded in Shapley value theory and weighted least squares regression, the method analytically derives a deterministic weight function, substantially reducing estimation variance. Furthermore, the implementation of KernelSHAP in the SHAP library is optimized to enhance computational simplicity and robustness while preserving theoretical consistency. Experiments demonstrate that, without compromising explanation fidelity, the approach reduces the number of contribution function evaluations by 5%–50% and accelerates inference by up to 50%, enabling real-time, interpretable analysis for high-dimensional features and large-scale predictions. The core innovation lies in the first principled replacement of stochastic kernel weights with a theory-driven deterministic weighting mechanism—achieving improved efficiency, numerical stability, and scalability.

Existing feature attribution methods for learning-to-rank (LTR) lack ranking-aware theoretical foundations, often yielding contradictory or counterintuitive results that undermine interpretability. Method: This paper introduces the first game-theoretic, axiomatized framework for ranking—formally specifying ranking-specific axioms including ranking consistency and efficiency—and derives Rank-SHAP, the first axiomatic extension of Shapley values to ranking tasks. Contribution/Results: We evaluate Rank-SHAP on MSLR-WEB30K and Istella with state-of-the-art LTR models (e.g., LambdaMART, DeepRank) and validate it via user studies. Results demonstrate significant improvements in attribution consistency and alignment with human judgment. Axiomatic analysis further reveals that most existing attribution methods violate fundamental ranking axioms. This work establishes the first rigorous, axiom-based foundation for explainable LTR.

This work addresses the computational intractability of standard SHAP due to its #P-hard complexity in feature attribution by integrating causal knowledge into the interpretability framework. The authors propose Asymmetric Shapley Values (ASV) grounded in causal graphs, leveraging equivalence classes derived from topological orderings of the causal structure. They establish, for the first time, a polynomial-time exact algorithm for computing ASV under rooted directed tree structures and further develop an efficient approximation algorithm applicable to arbitrary causal directed acyclic graphs (DAGs). Experimental results demonstrate that the proposed approach substantially improves computational efficiency on real-world causal structures while preserving high-quality explanation fidelity.

This work addresses the lack of theoretical foundations for feature attribution in multi-output predictive models, particularly the unresolved question of whether Shapley values should be computed independently for each output. By extending classical cooperative game axioms—efficiency, symmetry, dummy player, and additivity—to the vector-valued setting, the paper establishes a rigidity theorem: any attribution rule satisfying these axioms must decompose as a component-wise sum across individual outputs. This result formally justifies the necessity of output-wise SHAP explanations. Empirical validation on biomedical benchmarks demonstrates that this component-wise approach not only preserves interpretability consistency but also substantially improves computational efficiency during both training and deployment of multi-output models.

This work addresses the high computational complexity and poor scalability of SHAP in high-dimensional settings by proposing XAI–TU, a novel cooperative game-theoretic framework that integrates equal surplus and proportional allocation principles. Within this framework, the authors design ESENSC_rev2, a low-complexity feature attribution rule that satisfies the null player property and other key interpretability axioms. ESENSC_rev2 is the first SHAP alternative to offer both a polynomial-time closed-form solution and a rigorous axiomatic foundation. Experimental results across multiple tabular datasets demonstrate that ESENSC_rev2 efficiently approximates exact SHAP values with high accuracy, achieving a favorable balance between computational efficiency and theoretical guarantees while significantly improving scalability in high-dimensional scenarios.

This work addresses the challenge of efficiently estimating Shapley values in settings where coalition evaluations are computationally expensive and severely budget-constrained. It introduces, for the first time, Bayesian experimental design to this problem by proposing an adaptive sampling method that leverages a Gaussian process surrogate model. The approach strategically selects the most informative coalitions for evaluation by maximizing expected information gain. By exploiting the linearity of Shapley values and properties of elementary symmetric polynomials, the method reduces the computational complexity from exponential to polynomial in the number of players. Empirical results demonstrate that under tight evaluation budgets, the proposed algorithm substantially outperforms existing baselines across multiple high-cost application scenarios, achieving markedly higher sample efficiency.

Traditional Shapley value computation is computationally prohibitive, and existing learnable explanation methods struggle with the non-uniform grids and irregular geometries commonly encountered in physical simulations. This work proposes OperatorSHAP—the first mesh-agnostic attribution method that extends Shapley values to function spaces. By integrating neural operator architectures with a learnable explainer, OperatorSHAP delivers consistent explanations across varying mesh resolutions without requiring model retraining. The method establishes a theoretical connection to the Aumann–Shapley value and demonstrates strong empirical alignment with discrete Shapley values across multiple grid resolutions. Consequently, it significantly enhances both the efficiency and generalization of model interpretability in physics-informed applications.