Improved Convergence Rates for Distributed Resource Allocation

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 projectionbased algorithm which can handle smooth objective and simple constraints more efficiently. Numerical experiments demonstrates the viability and performance of all the proposed algorithms. DRAFT n i=1 (R i x i − r i ) ∈ K where K is a convex cone and R i is a matrix that couples local resources. It was required that the interior of K is nonempty which does not apply to (1b).The authors recently removed such nonempty interior requirement on K during our preparation of this paper. We also would like to point out that our rates are non-ergodic and the measures/criteria used for our rates have some advantages in practical uses (see the comments following Theorem 2). Based December 18, 2018 DRAFT

Distributed Dual Gradient Tracking for Resource Allocation in Unbalanced Networks

“…In addition, there are also several ADMM based methods that only work on balanced networks [7]- [9]. By exploiting the mirror relationship between the distributed optimization and distributed resource allocation, several accelerated distributed resource allocation algorithms are given in [10]. Moreover, the works [11] and [12] study continuous-time algorithms for DRAPs by using control theory tools.…”

“…To our best knowledge, these convergence results are only established for undirected balanced networks in [10]. Although a distributed algorithm for directed networks is also proposed in [10], there is no convergence result. We finally illustrate the advantages of DCGT over existing algorithms via simulation.…”

“…Since the resource allocation problem (1) is a constrained optimization problem, there are several ways to analyze the convergence rate. For example, [10] and [23] show convergence rates from a dual perspective based on the optimality condition (5), and n i,j=1 ∇F (w…”

“…Moreover, we further assume the local constraint set is the whole space, i.e., W i = R m . This is also assumed in [6], [10] and can be relaxed if the dual-based convergence rate evaluation is used as…”

“…This section validates our theoretical results and compares the DCGT with several state-of-the-art algorithms via simulation. Specifically, we compare the DCGT with the algorithms in [16], [23] and [10,. Note that [10] provides no convergence guarantee for Mirror-Push-DIGing, [16] has no convergence rate results, and [23] only show the convergence rate for strongly convex and Lipschitz smooth cost functions.…”