Federated learning is an emerging decentralized machine learning scheme that
allows multiple data owners to work collaboratively while ensuring data
privacy. The success of federated learning depends largely on the participation
of data owners. To sustain and encourage data owners' participation, it is
crucial to fairly evaluate the quality of the data provided by the data owners
and reward them correspondingly. Federated Shapley value, recently proposed by
Wang et al. [Federated Learning, 2020], is a measure for data value under the
framework of federated learning that satisfies many desired properties for data
valuation. However, there are still factors of potential unfairness in the
design of federated Shapley value because two data owners with the same local
data may not receive the same evaluation. We propose a new measure called
completed federated Shapley value to improve the fairness of federated Shapley
value. The design depends on completing a matrix consisting of all the possible
contributions by different subsets of the data owners. It is shown under mild
conditions that this matrix is approximately low-rank by leveraging concepts
and tools from optimization. Both theoretical analysis and empirical evaluation
verify that the proposed measure does improve fairness in many circumstances