Distributed optimization over time-varying directed graphs

TL;DR

This paper introduces the subgradient-push algorithm for distributed convex optimization over time-varying directed graphs, achieving convergence without prior knowledge of network size or topology, with a rate of O(ln(t)/√t).

Contribution

It proposes a novel broadcast-based algorithm that handles directed, time-varying communication graphs with no need for nodes to know the total number of agents or graph sequence.

Findings

Converges at a rate of O(ln(t)/√t)

Requires only out-degree knowledge at each node

Handles directed, time-varying graphs with strong connectivity

Abstract

We consider distributed optimization by a collection of nodes, each having access to its own convex function, whose collective goal is to minimize the sum of the functions. The communications between nodes are described by a time-varying sequence of directed graphs, which is uniformly strongly connected. For such communications, assuming that every node knows its out-degree, we develop a broadcast-based algorithm, termed the subgradient-push, which steers every node to an optimal value under a standard assumption of subgradient boundedness. The subgradient-push requires no knowledge of either the number of agents or the graph sequence to implement. Our analysis shows that the subgradient-push algorithm converges at a rate of $O(\ln(t)/\sqrt{t})$, where the constant depends on the initial values at the nodes, the subgradient norms, and, more interestingly, on both the consensus speed and the imbalances of influence among the nodes.