Asynchronous Distributed Optimization via Dual Decomposition and Block Coordinate Subgradient Methods

TL;DR

This paper introduces an asynchronous distributed optimization algorithm based on dual decomposition and block coordinate subgradient methods, capable of handling non-differentiable convex functions with overlapping dependencies, and provides convergence guarantees.

Contribution

It extends existing asynchronous optimization methods by allowing overlapping block updates, non-uniform probabilities, and local stepsizes without global coordination, with proven convergence properties.

Findings

Almost sure convergence under weaker assumptions

Sublinear convergence rate, linear under strong convexity

Effective in handling overlapping block updates

Abstract

We study the problem of minimizing the sum of potentially non-differentiable convex cost functions with partially overlapping dependences in an asynchronous manner, where communication in the network is not coordinated. We study the behavior of an asynchronous algorithm based on dual decomposition and block coordinate subgradient methods under assumptions weaker than those used in the literature. At the same time, we allow different agents to use local stepsizes with no global coordination. Sufficient conditions are provided for almost sure convergence to the solution of the optimization problem. Under additional assumptions, we establish a sublinear convergence rate that in turn can be strengthened to linear convergence rate if the problem is strongly convex and has Lipschitz gradients. We also extend available results in the literature by allowing multiple and potentially overlapping blocks to be updated at the same time with non-uniform and potentially time varying probabilities assigned to different blocks. A numerical example is provided to illustrate the effectiveness of the algorithm.