Data valuation: The partial ordinal Shapley value for machine learning

Data valuation using Shapley value has emerged as a prevalent research domain in machine learning applications. However, it is a challenge to address the role of order in data cooperation as most research lacks such discussion. To tackle this problem, this paper studies the definition of the partial ordinal Shapley value by group theory in abstract algebra. Besides, since the calculation of the partial ordinal Shapley value requires exponential time, this paper also gives three algorithms for approximating the results. The Truncated Monte Carlo algorithm is derived from the classic Shapley value approximation algorithm. The Classification Monte Carlo algorithm and the Classification Truncated Monte Carlo algorithm are based on the fact that the data points in the same class provide similar information, then we can accelerate the calculation by leaving out some data points in each class.