  Horizontal Federated learning (FL) handles multi-client data that share the
same set of features, and vertical FL trains a better predictor that combine
all the features from different clients. This paper targets solving vertical FL
in an asynchronous fashion, and develops a simple FL method. The new method
allows each client to run stochastic gradient algorithms without coordination
with other clients, so it is suitable for intermittent connectivity of clients.
This method further uses a new technique of perturbed local embedding to ensure
data privacy and improve communication efficiency. Theoretically, we present
the convergence rate and privacy level of our method for strongly convex,
nonconvex and even nonsmooth objectives separately. Empirically, we apply our
method to FL on various image and healthcare datasets. The results compare
favorably to centralized and synchronous FL methods.
