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

作者：Wei Shi, Qing Ling, Gang Wu, Wotao Yin · 发表于：SIAM Journal on Optimization · 年份：2015 · DOI：10.1137/14096668x · 被引用次数：1327 · 研究领域：Distributed Control Multi-Agent Systems、Neural Networks Stability and Synchronization、Energy Efficient Wireless Sensor Networks

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 $\mathrm{minimize}_{x\in\mathbb{R}^p}~\bar{f}(x)=\frac{1}{n}\sum_{i=1}^n f_i(x),$ 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 of $\bar{f}$. 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 per-iteration 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 a...