qshap: Fast Shapley Decomposition of $R^2$ for Gradient-Boosted Trees

Shapley values are a standard tool for explaining predictions of tree ensembles, with Path-Dependent SHAP being the most widely used variant. Despite substantial progress, existing methods still exhibit trade-offs between depth-dependent runtime, numerical stability, and support for higher-order interactions. To address these challenges, we introduce Quadrature-TreeSHAP, a quadrature-based reformulation of Path-Dependent TreeSHAP that is numerically stable, naturally extends to any-order Shapley interaction values and is practically insensitive to tree depth. Our implementation supports both CPU and GPU and is integrated into XGBoost. Our method is based on a weighted-Banzhaf interaction polynomial, which expresses Banzhaf interaction values as expectations under a feature participation probability $p$. Shapley values and any-order interaction values are then recovered by integrating these polynomials over $p$ from 0 to 1. We evaluate these integrals using Gauss-Legendre quadrature, and show that, in practice, only 8 fixed quadrature points are sufficient to reach machine precision. In fact, Quadrature-TreeSHAP with 8 fixed points achieves greater numerical stability than TreeSHAP. This fixed-point formulation removes depth dependence from the inner computation and enables efficient SIMD execution. We confirm these advantages empirically. On 12 XGBoost benchmarks, Quadrature-TreeSHAP computes Shapley values 1.06x-10.59x faster than TreeSHAP on CPU and 1.84x-6.95x faster than GPUTreeSHAP on GPU. Shapley pairwise interactions are 3.80x-58.11x faster on CPU, with higher-order interactions achieving speedups of up to 1200x compared to TreeSHAP-IQ.

Feature attribution analysis is critical for interpreting machine learning models and supporting reliable data-driven decisions. However, feature attribution measures often exhibit stochastic variation: different train--test splits, random seeds, or model-fitting procedures can produce substantially different attribution values and feature rankings. This paper proposes a framework for incorporating stochastic nature of feature attribution and a robust attribution metric, RoSHAP, for stable feature ranking based on the SHAP metric. The proposed framework models the distribution of feature attribution scores and estimates it through bootstrap resampling and kernel density estimation. We show that, under mild regularity conditions, the aggregated feature attribution score is asymptotically Gaussian, which greatly reduces the computational cost of distribution estimation. The RoSHAP summarizes the distribution of SHAP into a robust feature-ranking criterion that simultaneously rewards features that are active, strong, and stable. Through simulations and real-data experiments, the proposed framework and RoSHAP outperform standard single-run attribution measures in identifying signal features. In addition, models built using RoSHAP-selected features achieve predictive performance comparable to full-feature models while using substantially fewer predictors. The proposed RoSHAP approach improves the stability and interpretability of machine learning models, enabling reliable and consistent insights for analysis.

In this paper, we provide a theoretically grounded and computationally efficient alternative to SHAP. To this end, we study feature attribution through the lens of cooperative game theory by formulating a class of XAI--TU games. Building on this formulation, we investigate equal-surplus-type and proportional-allocation-type attribution rules and propose a low-cost attribution rule, ESENSC_rev2, constructed by combining two polynomial-time closed-form rules while ensuring the null-player property in the XAI--TU domain. Extensive experiments on tabular prediction tasks demonstrate that ESENSC_rev2 closely approximates exact SHAP while substantially improving scalability as the number of features increases. These empirical results indicate that equal-surplus-type attribution rules can achieve favorable trade-offs between computational cost and approximation accuracy in high-dimensional explainability settings. To provide theoretical foundations for these findings, we establish an axiomatic characterization showing that ESENSC_rev2 is uniquely determined by efficiency, the null-player axiom, a restricted differential marginality principle, an intermediate inessential-game property, and axioms that reduce computational requirements. Our results suggest that axiomatically justified and computationally efficient attribution rules can serve as practical and theoretically principled substitutes for SHAP-based approximations in modern explainability pipelines.