Horizontal and Vertical Federated Causal Structure Learning via Higher-order Cumulants

Federated causal discovery aims to uncover causal relationships while protecting data privacy, with significant real-world applications. Existing methods focus on horizontal federated settings where clients share the same variables but have different samples. However, in practice, clients may have different variables, leading to spurious causal relationships. To address this issue, we comprehensively consider causal structure learning methods under both horizontal and vertical federated settings. Interestingly, we find that, higher-order cumulants rely solely on the joint distribution of the relevant variables and are useful to solve the above problem in the linear non-Gaussian case. This motivates us to provide the identification theories for determining the causal order over observed variables, leveraging the difference in the product of the (cross) cumulants of the specific variables. Based on these theories, we develop a method for learning causal order in the horizontal and vertical federated scenarios. Specifically, we first obtain local (cross) cumulant matrices of observed variables from all participating clients to construct a global cumulant matrix. This global cumulant matrix is then used for recursive source variable identification, ultimately yielding a causal strength matrix of the union of variables from all clients. Our algorithm demonstrates superior performance in experiments on both synthetic and real-world data.