Blended Dynamics Approach to Distributed Optimization: Sum Convexity and Convergence Rate

TL;DR

This paper introduces a blended dynamics approach for distributed optimization that allows non-convex local functions, guarantees convergence close to centralized rates, and includes algorithms with minimal communication and flexible agent participation.

Contribution

It presents a novel blended dynamics framework enabling distributed optimization with non-convex local functions and near-centralized convergence rates, including new algorithms with flexible features.

Findings

Distributed algorithms achieve convergence rates close to centralized methods.

The approach allows non-convex local functions if the global function is strongly convex.

A distributed heavy-ball method demonstrates improved performance.

Abstract

This paper studies the application of the blended dynamics approach towards distributed optimization problem where the global cost function is given by a sum of local cost functions. The benefits include (i) individual cost function need not be convex as long as the global cost function is strongly convex and (ii) the convergence rate of the distributed algorithm is arbitrarily close to the convergence rate of the centralized one. Two particular continuous-time algorithms are presented using the proportional-integral-type couplings. One has benefit of `initialization-free,' so that agents can join or leave the network during the operation. The other one has the minimal amount of communication information. After presenting a general theorem that can be used for designing distributed algorithms, we particularly present a distributed heavy-ball method and discuss its strength over other methods.