Subgradient-Push Is of the Optimal Convergence Rate

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 $O\left( {\ln t/\sqrt t } \right)$. This paper shows that the subgradient-push algorithm actually converges at a rate of $O\left( {1/\sqrt t } \right)$, 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.