A stochastic averaging gradient algorithm with multi‐step communication for distributed optimization

Abstract

This paper studies distributed convex optimization problems over an undirected network where all nodes cooperate to minimize a sum of local objective functions. Each local objective function is further assumed to be an average of several convex instantaneous functions. By incorporating the stochastic averaging gradient into the distributed first-order primal-dual method, a stochastic averaging gradient algorithm with multi-step communication is proposed to solve the optimization problem. For each node, one randomly selected gradient of an instantaneous function is evaluated per iteration, which effectively reduces the computation cost of the algorithm. Based on conditions of strong convexity and Lipschitz continuity of the local instantaneous functions, a linear convergence rate of the proposed algorithm is guaranteed. Numerical simulations on the logistic regression problem demonstrate the performance of the algorithm and correctness of the theoretical results.