EXTRA: An Exact First-Order Algorithm for Decentralized Consensus Optimization

Abstract

Recently, there has been growing interest in solving consensus optimization problems in a multiagent network. In this paper, we develop a decentralized algorithm for the consensus optimization problem minimize x∈R pf (x) = 1 n n i=1 f i (x), which is defined over a connected network of n agents, where each function f i is held privately by agent i and encodes the agent's data and objective. All the agents shall collaboratively find the minimizer while each agent can only communicate with its neighbors. Such a computation scheme avoids a data fusion center or long-distance communication and offers better load balance to the network. This paper proposes a novel decentralized exact first-order algorithm (abbreviated as EXTRA) to solve the consensus optimization problem. "Exact" means that it can converge to the exact solution. EXTRA uses a fixed, large step size, which can be determined independently of the network size or topology. The local variable of every agent i converges uniformly and consensually to an exact minimizer off . In contrast, the well-known decentralized gradient descent (DGD) method must use diminishing step sizes in order to converge to an exact minimizer. EXTRA and DGD have the same choice of mixing matrices and similar periteration complexity. EXTRA, however, uses the gradients of the last two iterates, unlike DGD which uses just that of the last iterate. EXTRA has the best known convergence rates among the existing synchronized first-order decentralized algorithms for minimizing convex Lipschitz-differentiable functions. Specifically, if the f i 's are convex and have Lipschitz continuous gradients, EXTRA has an ergodic convergence rate O( 1 k ) in terms of the first-order optimality residual. In addition, as long asf is (restricted) strongly convex (not all individual f i 's need to be so), EXTRA converges to an optimal solution at a linear rate O(C −k ) for some constant C > 1. [9,22,26], estimation [1,2,16,18,29], sparse optimization [19,35], and low-rank matrix completion [20] problems. Functions f i can take the form of least squares [7,15,34], regularized least squares [1,2,9,18,22], as well as more general ones [26]. EXACT ALGORITHM FOR DECENTRALIZED OPTIMIZATION 945The solution x can represent, for example, the average temperature of a room [7, 34], frequency-domain occupancy of spectra [1, 2], states of a smart grid system [10, 16], sparse vectors [19,35], a matrix factor [20], and so on. In general, decentralized optimization fits the scenarios in which the data are collected and/or stored in a distributed network, a fusion center is either infeasible or not economical, and/or computing is required to be performed in a decentralized and collaborative manner by multiple agents. Related methods.Existing first-order decentralized methods for solving (1.1) include the (sub)gradient method [21,25,36], the (sub)gradient-push method [23,24], the fast (sub)gradient method [5,14], and the dual averaging method [8]. Compared to classical centralized algorithms, decentralized algorithms encoun...

Exact diffusion strategy for optimization by networked agents

“…The result in [9] shows that the EXTRA technique resolves the bias problem in consensus implementations. However, it is known that diffusion strategies outperform consensus strategies [3], [4].…”

“…In the work [9], a modified implementation of consensus iterations, referred to as EXTRA, was shown to remove the bias and converge to the exact minimizer w o rather than to an O(µ 2 )−neighborhood around w o . Motivated by [9], other variations with similar properties were proposed in [10], [11].…”

“…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

“…Clearly, PG-ExtraPush extends ExtraPush to handle nonsmooth objective terms. This extension is not the same as the extension from the gradient method to the proximal-gradient method, as well as the extension from EXTRA [15] to PG-EXTRA [16]. As the reader will see, PG-ExtraPush will have three interlaced sequences of iterates, whereas the proximal-gradient method just inherits of the sequence of iterates in the gradient method; and PG-ExtraPush uses the proximal maps of a sequence of transformed functions of r i associated with a positive weight sequence {w t } essentially introduced by the directed graph, while PG-EXTRA utilizes the proximity operator of r i .…”