Computing Exact Shapley Values in Polynomial Time for Product-Kernel Methods

2025-05-22

AI Summary

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.

Technology Category

• Machine Learning: Kernel Methods
• Philosophy and Ethics of AI: Accountability, Interpretability & Explainability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Kernel methods are widely used in machine learning due to their flexibility and expressive power. However, their black-box nature poses significant challenges to interpretability, limiting their adoption in high-stakes applications. Shapley value-based feature attribution techniques, such as SHAP and kernel-specific variants like RKHS-SHAP, offer a promising path toward explainability. Yet, computing exact Shapley values remains computationally intractable in general, motivating the development of various approximation schemes. In this work, we introduce PKeX-Shapley, a novel algorithm that utilizes the multiplicative structure of product kernels to enable the exact computation of Shapley values in polynomial time. We show that product-kernel models admit a functional decomposition that allows for a recursive formulation of Shapley values. This decomposition not only yields computational efficiency but also enhances interpretability in kernel-based learning. We also demonstrate how our framework can be generalized to explain kernel-based statistical discrepancies such as the Maximum Mean Discrepancy (MMD) and the Hilbert-Schmidt Independence Criterion (HSIC), thus offering new tools for interpretable statistical inference.

Problem

• Exact Shapley value computation is intractable for kernel methods
• Interpretability challenges limit kernel methods in high-stakes applications
• Current Shapley approximations lack efficiency for product-kernel structures

Innovation

• Exact Shapley computation via product kernels
• Recursive functional decomposition for efficiency
• Generalizes to kernel-based statistical discrepancies