Distributed Saddle Point Problems for Strongly Concave-Convex Functions

In this article, we propose GT-GDA , a distributed optimization method to solve saddle point problems of the form: ${\min _{\mathbf {x}} \max _{\mathbf {y}} \lbrace F(\mathbf x,\mathbf y) :=G(\mathbf x) + \langle \mathbf y, \overline{P} \mathbf x \rangle - H(\mathbf y) \rbrace }$ , where the functions $G(\cdot)$ , $H(\cdot)$ , and the coupling matrix $\overline{P}$ are distributed over a strongly connected network of nodes. GT-GDA is a first-order method that uses gradient tracking to eliminate the dissimilarity caused by heterogeneous data distribution among the nodes. In the most general form, GT-GDA includes a consensus over the local coupling matrices to achieve the optimal (unique) saddle point, however, at the expense of increased communication. To avoid this, we propose a more efficient variant GT-GDA-Lite that does not incur additional communication and analyze its convergence in various scenarios. We show that GT-GDA converges linearly to the unique saddle point solution when $G$ is smooth and convex, $H$ is smooth and strongly convex, and the global coupling matrix $\overline{P}$ has full column rank. We further characterize the regime under which GT-GDA exhibits a network topology-independent convergence behavior. We next show the linear convergence of GT-GDA-Lite to an error around the unique saddle point, which goes to zero when the coupling cost ${\langle \mathbf y, \overline{P} \mathbf x \rangle }$ is common to all nodes, or when $G$ and $H$ are quadratic. Numerical experiments illustrate the convergence properties and importance of GT-GDA and GT-GDA-Lite for several applications.