QuadraSHAP: Stable and Scalable Shapley Values for Product Games via Gauss-Legendre Quadrature

2026-05-07

AI Summary

This work proposes an efficient method for computing Shapley values in cooperative games with product-structured characteristic functions, commonly arising in interpretable machine learning models such as kernel methods and tree ensembles. By deriving the first exact one-dimensional integral representation of the Shapley value, the approach transforms the exponentially complex computation into a numerical integration task. High-accuracy approximations are achieved through Gauss–Legendre quadrature, numerically stable evaluation in log-space, and a parallelized correlated sampling algorithm. Theoretical analysis shows that the approximation error decays geometrically with the number of quadrature points, and experiments demonstrate that only a few hundred integration points suffice to yield the fastest and most numerically stable Shapley value estimates to date—even in settings with thousands of features.

Abstract

We study the efficient computation of Shapley values for \emph{product games} -- cooperative games in which the coalition value factorizes as a product of per-player terms. Such games arise in machine learning explainability whenever the value function inherits a multiplicative structure from the underlying model, as in kernel methods with product kernels and tree-based models. Our key result is that the Shapley value of each player in a product game admits an exact one-dimensional integral representation: the weighted sum over exponentially many feature coalitions collapses to the integral of a degree-$(d-1)$ polynomial over $[0,1]$, where $d$ is the total number of features. This yields a Gauss--Legendre quadrature scheme that is \emph{provably exact} whenever the number of nodes satisfies $m_q \geq \lceil d/2 \rceil$, and otherwise provides a \emph{near-exact} approximation with error provably decaying geometrically in $m_q$. In practice, a few hundred nodes can achieve highly precise estimates even with thousands of features. Building on this formulation, we derive a numerically stable implementation via log-space evaluation, together with an efficient parallel implementation based on associative scan primitives that achieves $O(d\,m_q)$ total work and $O(\log d)$ parallel time. Experiments show that \textsc{QuadraSHAP} is the fastest numerically stable method across all tested configurations.

Problem

• Shapley values
• product games
• machine learning explainability
• cooperative games
• multiplicative structure

Innovation

• Shapley values
• product games
• Gauss-Legendre quadrature
• numerical stability
• parallel algorithm