Distributed Optimization via Primal-Dual Gradient Dynamics with Stochastic Interactions

In this paper, we analyze the performance of the primal-dual gradient dynamics algorithm in the presence of stochastic communication channel uncertainty. In contrast to the existing results on the analysis of discretized primal-dual gradient dynamics with communication channel uncertainty, the main contribution of this work is in analyzing the stochastic continuous-time primal-dual gradient dynamics. Primal-dual gradient dynamics for distributed optimization are naturally modeled as a continuous-time dynamical system and analysis of this dynamics help us understand fundamental limitations and trade-offs between the cost function, network topology, and channel uncertainty for distributed optimization. We analyze the mean square stochastic stability of primal-dual gradient dynamics with communication channel uncertainty. The network topology is said to be more robust for distributed optimization if it can tolerate maximum variance of communication uncertainty. One of the important results of this paper is to show the existence of an optimal number of neighbors individual agent should have for robust distributed optimization. The optimal number is a function of network topology and cost function. If the network has more or less number of neighbors than the optimal number, then the network performance degrades. Simulation results involving nearest network topology are presented to verify the main conclusion of this paper.