Distributed Random Projection Algorithm for Convex Optimization

Random projection algorithm is an iterative gradient method with random projections. Such an algorithm is of interest for constrained optimization when the constraint set is not known in advance or the projection operation on the whole constraint set is computationally prohibitive. This paper presents a distributed random projection (DRP) algorithm for fully distributed constrained convex optimization problems that can be used by multiple agents connected over a time-varying network, where each agent has its own objective function and its own constrained set. With reasonable assumptions, we prove that the iterates of all agents converge to the same point in the optimal set almost surely. In addition, we consider a variant of the method that uses a mini-batch of consecutive random projections and establish its convergence in almost sure sense. Experiments on distributed support vector machines demonstrate fast convergence of the algorithm. It actually shows that the number of iteration required until convergence is much smaller than scanning over all training samples just once.