Shapley Value Estimation based on Differential Matrix

The Shapley value has been extensively used in many fields as the unique metric to fairly evaluate player contributions in cooperative settings. Since the exact computation of Shapley values is #P-hard in the task-agnostic setting, many studies have been developed to utilize the Monte Carlo method for Shapley value estimation. The existing methods estimate the Shapley values directly. In this paper, we explore a novel idea-inferring the Shapley values by estimating the differences between them. Technically, we estimate a differential matrix consisting of pairwise Shapley value differences to reduce the variance of the estimated Shapley values. We develop a least-squares optimization solution to derive the Shapley values from the differential matrix, minimizing the estimator variances. Additionally, we devise a Monte Carlo method for efficient estimation of the differential matrix and introduce two stratified Monte Carlo methods for further variance reduction. Our experimental results on real and synthetic data sets demonstrate the effectiveness and efficiency of the differential-matrix-based sampling approaches.