Almost sure convergence of distributed optimization with imperfect information sharing

We study convex distributed optimization problems where a set of agents are interested in solving a separable optimization problem collaboratively with imperfect information sharing over time-varying networks. A robust algorithm is designed to reduce communication load and meet rate constraints. We study the almost sure convergence of a two-time-scale decentralized gradient descent algorithm to reach the consensus on an optimizer of the objective loss function. One time scale fades out the imperfect incoming information from neighboring agents, and the second one adjusts the local loss functions’ gradients. We show that under certain conditions on the connectivity of the underlying time-varying network and the time-scale sequences, the dynamics converge almost surely to an optimal point supported in the optimizer set of the loss function.