Discretized Distributed Optimization over Dynamic Digraphs

TL;DR

This paper introduces a discretized distributed optimization algorithm for dynamic directed graphs that does not require stochastic weight design, enabling robust distributed learning over volatile networks.

Contribution

It presents a novel optimization method that operates over general strongly connected dynamic networks without the need for bi-stochastic weight design, simplifying implementation in changing topologies.

Findings

Convergence is guaranteed with specific gradient-tracking step-size bounds.

The method is robust to link failures and packet drops.

It improves over existing stochastic-weight approaches in dynamic networks.

Abstract

We consider a discrete-time model of continuous-time distributed optimization over dynamic directed-graphs (digraphs) with applications to distributed learning. Our optimization algorithm works over general strongly connected dynamic networks under switching topologies, e.g., in mobile multi-agent systems and volatile networks due to link failures. Compared to many existing lines of work, there is no need for bi-stochastic weight designs on the links. The existing literature mostly needs the link weights to be stochastic using specific weight-design algorithms needed both at the initialization and at all times when the topology of the network changes. This paper eliminates the need for such algorithms and paves the way for distributed optimization over time-varying digraphs. We derive the bound on the gradient-tracking step-size and discrete time-step for convergence and prove dynamic stability using arguments from consensus algorithms, matrix perturbation theory, and Lyapunov theory. This work, particularly, is an improvement over existing stochastic-weight undirected networks in case of link removal or packet drops. This is because the existing literature may need to rerun time-consuming and computationally complex algorithms for stochastic design, while the proposed strategy works as long as the underlying network is weight-symmetric and balanced. The proposed optimization framework finds applications to distributed classification and learning.