Vertical federated learning (VFL) is a promising approach for collaboratively training machine learning models using private data partitioned vertically across different parties. Ideally in a VFL setting, the active party (party possessing features of samples with labels) benefits by improving its machine learning model through collaboration with some passive parties (parties possessing additional features of the same samples without labels) in a privacy preserving manner. However, motivating passive parties to participate in VFL can be challenging. In this paper, we focus on the problem of allocating incentives to the passive parties by the active party based on their contributions to the VFL process. We formulate this problem as a variant of the Nucleolus game theory concept, known as the Bankruptcy Problem, and solve it using the Talmud's division rule. We evaluate our proposed method on synthetic and real-world datasets and show that it ensures fairness and stability in incentive allocation among passive parties who contribute their data to the federated model. Additionally, we compare our method to the existing solution of calculating Shapley values and show that our approach provides a more efficient solution with fewer computations.

翻译：纵向联邦学习（VFL）是一种利用各参与方垂直划分的私有数据协同训练机器学习模型的有效方法。在理想情况下，主动方（拥有带标签样本特征的参与方）通过与若干被动方（拥有相同样本附加特征但无标签的参与方）进行隐私保护协作，可提升其机器学习模型性能。然而，激励被动方参与VFL具有挑战性。本文聚焦于主动方根据被动方对VFL过程的贡献分配激励机制的问题。我们将其建模为核仁博弈理论概念的变体——破产问题，并采用《塔木德》分配法则求解。通过在合成数据集与真实数据集上的实验评估，证明该方法能确保贡献数据给联邦模型的被动方获得公平且稳定的激励机制分配。此外，与现有基于夏普利值的计算方法相比，本方法以更少的计算量实现了更高效的解决方案。