  The push-sum based subgradient is an important method for distributed convex
optimization over unbalanced directed graphs, which is known to converge at a
rate of $O(\ln t/\sqrt{t})$. This paper shows that the subgradient-push
algorithm actually converges at a rate of $O(1/\sqrt{t})$, which is the same as
that of the single-agent subgradient and thus optimal. The proposed tool for
analyzing push-sum based algorithms is of independent interest.
