  Federated learning has become a popular tool in the big data era nowadays. It
trains a centralized model based on data from different clients while keeping
data decentralized. In this paper, we propose a federated sparse sliced inverse
regression algorithm for the first time. Our method can simultaneously estimate
the central dimension reduction subspace and perform variable selection in a
federated setting. We transform this federated high-dimensional sparse sliced
inverse regression problem into a convex optimization problem by constructing
the covariance matrix safely and losslessly. We then use a linearized
alternating direction method of multipliers algorithm to estimate the central
subspace. We also give approaches of Bayesian information criterion and
hold-out validation to ascertain the dimension of the central subspace and the
hyper-parameter of the algorithm. We establish an upper bound of the
statistical error rate of our estimator under the heterogeneous setting. We
demonstrate the effectiveness of our method through simulations and real world
applications.
