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Recently, there have been growing interests in solving consensus optimizationproblems in a multi-agent network. In this paper, we develop a decentralizedalgorithm for the consensus optimization problem$$\min\limits_{x\in\mathbb{R}^p}~\bar{f}(x)=\frac{1}{n}\sum\limits_{i=1}^nf_i(x),$$ which is defined over a connected network of $n$ agents, where eachfunction $f_i$ is held privately by agent $i$ and encodes the agent's data andobjective. All the agents shall collaboratively find the minimizer while eachagent can only communicate with its neighbors. Such a computation scheme avoidsa data fusion center or long-distance communication and offers better loadbalance 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 can use a fixed largestep size, {which is independent of the network size}, and has synchronizediterations. The local variable of every agent $i$ converges uniformly andconsensually to an exact minimizer of $\bar{f}$. In contrast, the well-knowndecentralized gradient descent (DGD) method must use diminishing step sizes inorder to converge to an exact minimizer. EXTRA and DGD have the same choice ofmixing matrices and similar per-iteration complexity. EXTRA, however, uses thegradients of last two iterates, unlike DGD which uses just that of lastiterate.  EXTRA has the best known convergence rates among the existing first-orderdecentralized algorithms. Specifically, if $f_i$'s are convex and haveLipschitz continuous gradients, EXTRA has an ergodic convergence rate$O(\frac{1}{k})$ in terms of the first-order optimality residual. If $\bar{f}$is also restricted strongly convex, EXTRA converges to an optimal solution at alinear rate $O(C^{-k})$ for some constant $C>1$.

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