On the convergence of the distributed proximal point algorithm

In this work, we establish convergence results for the distributed proximal point algorithm (DPPA) for distributed optimization problems. We consider the problem on the whole domain Rd and find a general condition on the stepsize and cost functions such that the DPPA is stable. We prove that the DPPA with stepsize $\eta > 0$ exponentially converges to an $O(\eta)$-neighborhood of the optimizer. Our result clearly explains the advantage of the DPPA with respect to the convergence and stability in comparison with the distributed gradient descent algorithm. We also provide numerical tests supporting the theoretical results.

On the convergence result of the gradient-push algorithm on directed graphs with constant stepsize

Gradient-push algorithm has been widely used for decentralized optimization problems when the connectivity network is a direct graph. This paper shows that the gradient-push algorithm with stepsize $\alpha>0$ converges exponentially fast to an $O(\alpha)$-neighborhood of the optimizer under the assumption that each cost is smooth and the total cost is strongly convex. Numerical experiments are provided to support the theoretical convergence results.