Distributed deterministic asynchronous algorithms in time-varying graphs through Dykstra splitting

TL;DR

This paper introduces a distributed, deterministic asynchronous algorithm based on Dykstra's method for optimizing sum functions over time-varying graphs, ensuring convergence without central control.

Contribution

It develops a novel distributed Dykstra's algorithm that guarantees deterministic convergence in asynchronous, time-varying network settings, including accelerated and non-strongly convex cases.

Findings

Convergence guaranteed under connectivity conditions.

Algorithm operates asynchronously without a global clock.

Applicable to non-strongly convex functions.

Abstract

Consider the setting where each vertex of a graph has a function, and communications can only occur between vertices connected by an edge. We wish to minimize the sum of these functions. For the case when each function is the sum of a strongly convex quadratic and a convex function, we propose a distributed version of Dykstra's algorithm. The computations to optimize the dual objective function can run asynchronously without a global clock, and in a distributed manner without a central controller. Convergence to the primal minimizer is deterministic instead of being probabilistic, and is guaranteed as long as in each cycle, the edges where two-way communications occur connects all vertices. We also look at an accelerated algorithm, and an algorithm for the case when the functions on the nodes are not strongly convex.