Subgradient-Push Is of the Optimal Convergence Rate

TL;DR

This paper proves that the subgradient-push algorithm for distributed convex optimization over directed graphs converges at the optimal rate of O(1/√t), matching single-agent subgradient methods.

Contribution

It establishes the optimal convergence rate of the subgradient-push algorithm and introduces a new analytical tool for push-sum based algorithms.

Findings

Subgradient-push converges at O(1/√t) rate.

The convergence rate matches that of single-agent subgradient methods.

The analysis tool is of independent interest.

Abstract

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.