  This paper focuses on the distributed optimization of stochastic saddle point
problems. The first part of the paper is devoted to lower bounds for the
cenralized and decentralized distributed methods for smooth (strongly)
convex-(strongly) concave saddle-point problems as well as the near-optimal
algorithms by which these bounds are achieved. Next, we present a new federated
algorithm for cenralized distributed saddle point problems - Extra Step Local
SGD. Theoretical analysis of the new method is carried out for strongly
convex-strongly concave and non-convex-non-concave problems. In the
experimental part of the paper, we show the effectiveness of our method in
practice. In particular, we train GANs in a distributed manner.
