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

TL;DR

QuadraSHAP introduces a novel, efficient quadrature-based method for computing Shapley values in product games, significantly improving speed and stability in machine learning explainability tasks.

Contribution

It provides an exact integral representation of Shapley values in product games and develops a scalable, numerically stable quadrature scheme with provable accuracy.

Findings

Gauss-Legendre quadrature yields exact or near-exact Shapley value estimates.

A parallel implementation achieves O(d m_q) work and O(log d) time.

QuadraSHAP outperforms existing methods in speed and stability.

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.