Projected Push-Pull For Distributed Constrained Optimization Over Time-Varying Directed Graphs (extended version)

TL;DR

This paper presents the Projected Push-Pull algorithm for distributed constrained optimization over dynamic directed graphs, combining projected gradient descent and lazy updates to ensure convergence and constraint satisfaction.

Contribution

It introduces a novel algorithm that guarantees geometric convergence in distributed constrained optimization with time-varying directed graphs, incorporating lazy updates and explicit step size bounds.

Findings

Achieves geometric convergence over time-varying directed graphs.

Ensures decision variables remain within the constraint set.

Validated through numerical experiments on different graph types.

Abstract

We introduce the Projected Push-Pull algorithm that enables multiple agents to solve a distributed constrained optimization problem with private cost functions and global constraints, in a collaborative manner. Our algorithm employs projected gradient descent to deal with constraints and a lazy update rule to control the trade-off between the consensus and optimization steps in the protocol. We prove that our algorithm achieves geometric convergence over time-varying directed graphs while ensuring that the decision variable always stays within the constraint set. We derive explicit bounds for step sizes that guarantee geometric convergence based on the strong-convexity and smoothness of cost functions, and graph properties. Moreover, we provide additional theoretical results on the usefulness of lazy updates, revealing the challenges in the analysis of any gradient tracking method that uses projection operators in a distributed constrained optimization setting. We validate our theoretical results with numerical studies over different graph types, showing that our algorithm achieves geometric convergence empirically.