This paper considers the problem of decentralized consensus optimization over a network, where each node holds a strongly convex and twice-differentiable local objective function. Our goal is to minimize the sum of the local objective functions and find the exact optimal solution using only local computation and neighboring communication. We propose a novel Newton tracking algorithm, which updates the local variable in each node along a local Newton direction modified with neighboring and historical information. We investigate the connections between the proposed Newton tracking algorithm and several existing methods, including gradient tracking and primal-dual methods. We prove that the proposed algorithm converges to the exact optimal solution at a linear rate. Furthermore, when the iterate is close to the optimal solution, we show that the proposed algorithm requires $O(\max \lbrace \kappa _f \sqrt{\kappa _g} + \kappa _f^2, \frac{\kappa _g^{3/2}}{\kappa _f} + \kappa _f\sqrt{\kappa _g} \rbrace \log {\frac{1}{\Delta }})$ iterations to find a $\Delta$ -optimal solution, where $\kappa _f$ and $\kappa _g$ are condition numbers of the objective function and the graph, respectively. Our numerical results demonstrate the efficacy of Newton tracking and validate the theoretical findings.

中文翻译：

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一种用于去中心化共识优化的精确线性收敛的牛顿跟踪算法


本文考虑了网络上的去中心化共识优化问题，其中每个节点都拥有一个强凸且可二次微分的局部目标函数。我们的目标是最小化局部目标函数的总和，并仅使用局部计算和邻近通信找到精确的最优解。我们提出了一种新颖的牛顿跟踪算法，该算法沿着用邻近和历史信息修改的局部牛顿方向更新每个节点中的局部变量。我们研究了所提出的牛顿跟踪算法与几种现有方法（包括梯度跟踪和原始对偶方法）之间的联系。我们证明所提出的算法以线性速率收敛到精确的最优解。此外，当迭代接近最优解时，我们表明所提出的算法需要 $O(\max \lbrace \kappa _f \sqrt{\kappa _g} + \kappa _f^2, \frac{\kappa _g^ {3/2}}{\kappa _f} + \kappa _f\sqrt{\kappa _g} \rbrace \log {\frac{1}{\Delta }})$ 次迭代以找到 $\Delta$ 最优解，其中 $\kappa _f$ 和 $\kappa _g$ 分别是目标函数和图的条件数。我们的数值结果证明了牛顿跟踪的有效性并验证了理论结果。