  To design algorithms that reduce communication cost or meet rate constraints
and are robust to communication noise, 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. 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.
