Improved Convergence Rates for Distributed Resource Allocation

TL;DR

This paper introduces new decentralized algorithms for convex resource allocation problems in networks, achieving improved convergence rates and scalability, with applications to various convexity and constraint scenarios.

Contribution

The paper presents novel decentralized algorithms with proven convergence rates for resource allocation, including methods for convex, strongly convex, and constrained cases.

Findings

Algorithms achieve $o(1/k)$ convergence for convex objectives.

Gradient-based method attains geometric convergence for strongly convex functions.

Proposed methods demonstrate scalability and efficiency in numerical experiments.

Abstract

In this paper, we develop a class of decentralized algorithms for solving a convex resource allocation problem in a network of $n$ agents, where the agent objectives are decoupled while the resource constraints are coupled. The agents communicate over a connected undirected graph, and they want to collaboratively determine a solution to the overall network problem, while each agent only communicates with its neighbors. We first study the connection between the decentralized resource allocation problem and the decentralized consensus optimization problem. Then, using a class of algorithms for solving consensus optimization problems, we propose a novel class of decentralized schemes for solving resource allocation problems in a distributed manner. Specifically, we first propose an algorithm for solving the resource allocation problem with an $o(1/k)$ convergence rate guarantee when the agents' objective functions are generally convex (could be nondifferentiable) and per agent local convex constraints are allowed; We then propose a gradient-based algorithm for solving the resource allocation problem when per agent local constraints are absent and show that such scheme can achieve geometric rate when the objective functions are strongly convex and have Lipschitz continuous gradients. We have also provided scalability/network dependency analysis. Based on these two algorithms, we have further proposed a gradient projection-based algorithm which can handle smooth objective and simple constraints more efficiently. Numerical experiments demonstrates the viability and performance of all the proposed algorithms.