  We study the so-called distributed two-time-scale gradient method for solving
convex optimization problems over a network of agents when the communication
bandwidth between the nodes is limited, and so information that is exchanged
between the nodes must be quantized. Our main contribution is to provide a
novel analysis, resulting to an improved convergence rate of this method as
compared to the existing works. In particular, we show that the method
converges at a rate $O(log_2 k/\sqrt k)$ to the optimal solution, when the
underlying objective function is strongly convex and smooth. The key technique
in our analysis is to consider a Lyapunov function that simultaneously captures
the coupling of the consensus and optimality errors generated by the method.
