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.

Data valuation is a core function in data markets and cooperative data sharing.Shapley valueis a widely used approach to fairly measure the contribution of data points towards a collective utility (e.g., a machine learning model trained from the data). However, computing Shapley values is known to be in general #P-hard due to the exponential utility evaluation. Furthermore, the presence of dynamic data poses additional challenges due to the prohibitively expensive cost of recomputing from scratch. In this paper, we study the problem ofDynamic Shapley Value Computation, which focuses on updating Shapley values when dynamically adding or deleting data points. For adding, to prune redundant computation of overlapping model utilities, we propose the pivot-based algorithm that can reduce half the computation time in expectation. We also propose delta-based algorithms to capture Shapley value changes, which require only a smaller sample size to converge. For deleting, we present the YN-NN algorithm that derives the new Shapley values from precomputed utilities efficiently. Based on Shapley value changes, we give another version of the delta-based algorithm for deleting data points. Besides, we propose heuristic algorithms that draw on experimental observations for addition, deletion, and hybrid scenarios. Extensive experimental results demonstrate the efficiency and effectiveness of our proposed algorithms.

The Shapley value provides a unique approach to equitably gauge each player's contribution within a coalition and has extensive applications with various utility functions. In data valuation for machine learning, particularly for classification tasks, using classification accuracy as the utility function has become a de facto standard. However, accuracy can be an imprecise metric, potentially missing finer details crucial for valuation. In this paper, we propose the probability-based Shapley (P-Shapley) value, which leverages predicted probabilities to heighten utility differentiation. Several convex calibration functions are further incorporated for probability calibration. We prove that the P-Shapley value outperforms Shapley values based on accuracy or other coarse metrics in approximation stability and the discrimination of marginal utility change can be further improved by convex calibration functions. Extensive experiments on four real-world datasets demonstrate the effectiveness of our approaches.

With the prevalence of data-driven research, data valuation has attracted attention from the computer science field. How to appraise a single datum becomes an imperative problem, especially in the context of machine learning. Shapley value is widely used to fairly measure the contribution of data points in machine learning since it is the unique definition that satisfies all four desired properties: balance, symmetry, additivity, and zero element. However, computing Shapley value is known to be a #P-hard problem. As data is subject to changes, dynamic data exists pervasively in real-world scenarios. Pricing such dynamic data is more challenging due to the prohibitively expensive cost of recalculation from scratch. In this paper, we study the problem of Dynamic Shapley Value Computation, which updates Shapley value when dynamically adding/deleting data points. For adding data points, to prune unnecessary computation of overlapping model utilities, we propose the pivot-based algorithm that can reduce half computation time in general. We also propose the delta-based algorithm to capture Shapley value changes, which requires a smaller sample size to converge. For deleting data points, we present the YN-NN algorithm that derives the new Shapley value from the data structure of precomputed model utilities in an efficient way. Based on Shapley value changes, we give another version of the delta-based algorithm for deleting data points. Besides, we propose heuristic algorithms to draw on experimental observations for both adding and deleting data points. Extensive experimental results demonstrate the efficiency and effectiveness of our proposed algorithms.

The increasing demand for data-driven machine learning (ML) models has led to the emergence of model markets, where a broker collects personal data from data owners to produce high-usability ML models. To incentivize data owners to share their data, the broker needs to price data appropriately while protecting their privacy. Forequitable data valuation, which is crucial in data pricing,Shapley valuehas become the most prevalent technique because it satisfies all four desirable properties in fairness: balance, symmetry, zero element, and additivity. Forthe right to be forgotten, which is stipulated by many data privacy protection laws to allow data owners to unlearn their data from trained models, thesharded structurein ML model training has become a de facto standard to reduce the cost of future unlearning by avoiding retraining the entire model from scratch. In this paper, we explore how the sharded structure for the right to be forgotten affects Shapley value for equitable data valuation in model markets. To adapt Shapley value for the sharded structure, we propose S-Shapley value, a sharded structure-based Shapley value, which satisfies four desirable properties for data valuation. Since we prove that computing S-Shapley value is #P-complete, two sampling-based methods are developed to approximate S-Shapley value. Furthermore, to efficiently update valuation results after data owners unlearn their data, we present two delta-based algorithms that estimate the change of data value instead of the data value itself. Experimental results demonstrate the efficiency and effectiveness of the proposed algorithms.