  Federated learning is a popular technology for training machine learning
models on distributed data sources without sharing data. Vertical federated
learning or feature-based federated learning applies to the cases that
different data sources share the same sample ID space but differ in feature
space. To ensure the data owners' long-term engagement, it is critical to
objectively assess the contribution from each data source and recompense them
accordingly. The Shapley value (SV) is a provably fair contribution valuation
metric originated from cooperative game theory. However, computing the SV
requires extensively retraining the model on each subset of data sources, which
causes prohibitively high communication costs in federated learning. We propose
a contribution valuation metric called vertical federated Shapley value
(VerFedSV) based on SV. We show that VerFedSV not only satisfies many desirable
properties for fairness but is also efficient to compute, and can be adapted to
both synchronous and asynchronous vertical federated learning algorithms. Both
theoretical analysis and extensive experimental results verify the fairness,
efficiency, and adaptability of VerFedSV.
