Linear and sublinear convergence rates for a subdifferentiable distributed deterministic asynchronous Dykstra's algorithm

TL;DR

This paper establishes sublinear convergence for a distributed asynchronous Dykstra's algorithm and shows linear convergence under smoothness assumptions, advancing understanding of convergence behavior in distributed optimization.

Contribution

It provides the first convergence rate analysis for a distributed asynchronous Dykstra's algorithm applied to subdifferentiable functions, including linear rates for smooth cases.

Findings

Proves sublinear convergence rates for the general algorithm.

Establishes linear convergence when functions are smooth with Lipschitz gradients.

Extends previous work by analyzing convergence in a distributed asynchronous setting.

Abstract

In two earlier papers, we designed a distributed deterministic asynchronous algorithm for minimizing the sum of subdifferentiable and proximable functions and a regularizing quadratic on time-varying graphs based on Dykstra's algorithm, or block coordinate dual ascent. Each node in the distributed optimization problem is the sum of a known regularizing quadratic and a function to be minimized. In this paper, we prove sublinear convergence rates for the general algorithm, and a linear rate of convergence if the function on each node is smooth with Lipschitz gradient.