  Identifying functional connectivity biomarkers of major depressive disorder
(MDD) patients is essential to advance understanding of the disorder mechanisms
and early intervention. However, due to the small sample size and the high
dimension of available neuroimaging data, the performance of existing methods
is often limited. Multi-site data could enhance the statistical power and
sample size, while they are often subject to inter-site heterogeneity and
data-sharing policies. In this paper, we propose a federated joint estimator,
NOTEARS-PFL, for simultaneous learning of multiple Bayesian networks (BNs) with
continuous optimization, to identify disease-induced alterations in MDD
patients. We incorporate information shared between sites and site-specific
information into the proposed federated learning framework to learn
personalized BN structures by introducing the group fused lasso penalty. We
develop the alternating direction method of multipliers, where in the local
update step, the neuroimaging data is processed at each local site. Then the
learned network structures are transmitted to the center for the global update.
In particular, we derive a closed-form expression for the local update step and
use the iterative proximal projection method to deal with the group fused lasso
penalty in the global update step. We evaluate the performance of the proposed
method on both synthetic and real-world multi-site rs-fMRI datasets. The
results suggest that the proposed NOTEARS-PFL yields superior effectiveness and
accuracy than the comparable methods.
