Achieving Geometric Convergence for Distributed Optimization Over Time-Varying Graphs

Abstract

This paper considers the problem of distributed optimization over time-varying graphs. For the case of undirected graphs, we introduce a distributed algorithm, referred to as DIGing, based on a combination of a distributed inexact gradient method and a gradient tracking technique. The DIGing algorithm uses doubly stochastic mixing matrices and employs fixed step-sizes and, yet, drives all the agents' iterates to a global and consensual minimizer. When the graphs are directed, in which case the implementation of doubly stochastic mixing matrices is unrealistic, we construct an algorithm that incorporates the push-sum protocol into the DIGing structure, thus obtaining Push-DIGing algorithm. The Push-DIGing uses column stochastic matrices and fixed step-sizes, but it still converges to a global and consensual minimizer. Under the strong convexity assumption, we prove that the algorithms converge at R-linear (geometric) rates as long as the step-sizes do not exceed some upper bounds. We establish explicit estimates for the convergence rates. When the graph is undirected it shows that DIGing scales polynomially in the number of agents. We also provide some numerical experiments to demonstrate the efficacy of the proposed algorithms and to validate our theoretical findings.

Exact diffusion strategy for optimization by networked agents

“…In the second experiment, we compare exact diffusion with EXTRA consensus [9], DIGing [10], and Aug-DGM [11]. These algorithms require symmetric doubly-stochastic matrices or right-stochastic matrices.…”

“…Motivated by [9], other variations with similar properties were proposed in [10], [11]. These variations, compared to EXTRA, have two information combinations per recursion, which can be a burden when communication resources are limited.…”

“…These variations, compared to EXTRA, have two information combinations per recursion, which can be a burden when communication resources are limited. Moreover, while EXTRA [9] and ADMM-based algorithms [6], [7] require symmetric and doubly-stochastic combination matrices, the variations DIGing [10], ExtraPush [12], and Aug-DGM [11] require right-stochastic combination matrices. All these types of combination matrices impose stringent constraints on the network topology and communication protocols because each agent will need to be aware of its neighbors in advance.…”

A fast proximal gradient algorithm for decentralized composite optimization over directed networks

“…In the column-stochastic setting, the push-sum protocol [4] can be used to obtain a stationary distribution for the mixing matrix. Some recent decentralized algorithms over a directed network include Subgradient-Push [10], ExtraPush [22] (also called DEXTRA in [20]) and Push-DIGing [11]. The best rate of Subgradient-Push in the general convex case is O(ln t/ √ t), where t is the iteration number, and both ExtraPush and Push-DIGing perform linearly convergent in the strongly convex case.…”

A General Framework of Exact Primal-Dual First-Order Algorithms for Distributed Optimization

“…The authors in [36,32] introduced an exact first-order algorithm to solve the consensus optimization problem and established sublinear and linear rate of convergence under the convexity and strong convexity assumptions. The algorithm presented in [28] considers solving consensus problem over time varying and directed graphs and establishes linear rate of convergence for strongly convex functions. Although these algorithms do not involve dual variables explicitly, they can be viewed as primal-dual methods, which replace the primal minimization problem with a single gradient descent step.…”