Primal-Dual Stochastic Distributed Algorithm for Constrained Convex Optimization

This paper investigates distributed convex optimization problems over an undirected and connected network, where each node’s variable lies in a private constrained convex set, and overall nodes aim at collectively minimizing the sum of all local objective functions. Motivated by a variety of applications in machine learning problems with large-scale training sets distributed to multiple autonomous nodes, each local objective function is further designed as the average of moderate number of local instantaneous functions. Each local objective function and constrained set cannot be shared with others. A primal-dual stochastic algorithm is presented to address the distributed convex optimization problems, where each node updates its state by resorting to unbiased stochastic averaging gradients and projects on its private constrained set. At each iteration, for each node the gradient of one local instantaneous function selected randomly is evaluated and the average of the most recent stochastic gradients is used to approximate the true local gradient. In the constrained case, we show that with strong-convexity of the local instantaneous function and Lipschitz continuity of its gradient, the algorithm converges to the global optimization solution almost surely. In the unconstrained case, an explicit linear convergence rate of the algorithm is provided. Numerical experiments are presented to demonstrate correctness of the theoretical results.