Distributed Optimization: Convergence Conditions from a Dynamical System Perspective

TL;DR

This paper investigates the conditions under which distributed optimization algorithms achieve global optimal consensus, emphasizing the importance of the intersection of local solution sets and control laws for fixed and time-varying graphs.

Contribution

It establishes necessary and sufficient conditions for convergence in distributed optimization from a dynamical systems perspective, including error bounds and graph connectivity requirements.

Findings

Existence of control laws guaranteeing global optimal consensus under intersection conditions.

Approximate consensus achievable with mild conditions and error tolerance.

Optimal consensus can be achieved in time-varying graphs with joint strong connectivity.

Abstract

This paper explores the fundamental properties of distributed minimization of a sum of functions with each function only known to one node, and a pre-specified level of node knowledge and computational capacity. We define the optimization information each node receives from its objective function, the neighboring information each node receives from its neighbors, and the computational capacity each node can take advantage of in controlling its state. It is proven that there exist a neighboring information way and a control law that guarantee global optimal consensus if and only if the solution sets of the local objective functions admit a nonempty intersection set for fixed strongly connected graphs. Then we show that for any tolerated error, we can find a control law that guarantees global optimal consensus within this error for fixed, bidirectional, and connected graphs under mild conditions. For time-varying graphs, we show that optimal consensus can always be achieved as long as the graph is uniformly jointly strongly connected and the nonempty intersection condition holds. The results illustrate that nonempty intersection for the local optimal solution sets is a critical condition for successful distributed optimization for a large class of algorithms.