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Cooperative Convex Optimization in Networked Systems: Augmented Lagrangian Algorithms With Directed Gossip Communication

机译:网络系统中的合作凸优化:具有直接八卦通信的增强拉格朗日算法

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We study distributed optimization in networked systems, where nodes cooperate to find the optimal quantity of common interest, $x=x^{star}$. The objective function of the corresponding optimization problem is the sum of private (known only by a node), convex, nodes' objectives and each node imposes a private convex constraint on the allowed values of $x$ . We solve this problem for generic connected network topologies with asymmetric random link failures with a novel distributed, decentralized algorithm. We refer to this algorithm as AL–G (augmented Lagrangian gossiping), and to its variants as AL–MG (augmented Lagrangian multi neighbor gossiping) and AL–BG (augmented Lagrangian broadcast gossiping). The AL–G algorithm is based on the augmented Lagrangian dual function. Dual variables are updated by the standard method of multipliers, at a slow time scale. To update the primal variables, we propose a novel, Gauss-Seidel type, randomized algorithm, at a fast time scale. AL–G uses unidirectional gossip communication, only between immediate neighbors in the network and is resilient to random link failures. For networks with reliable communication (i.e., no failures), the simplified, AL–BG (augmented Lagrangian broadcast gossiping) algorithm reduces communication, computation and data storage cost. We prove convergence for all proposed algorithms and demonstrate by simulations the effectiveness on two applications: $l_{1}$–regularized logistic regression for classification and cooperative spectrum sensing for cognitive radio networks.
机译:我们研究网络系统中的分布式优化,其中节点协作以找到共同感兴趣的最佳数量 $ x = x ^ {star} $ < / formula>。相应优化问题的目标函数是私有(仅由节点知道),凸面,节点目标的总和,并且每个节点对 $ x $ 。我们通过一种新颖的分布式分散算法,为具有非对称随机链路故障的通用连接网络拓扑解决了该问题。我们将此算法称为AL–G(增强型Lagrangian <?Pub / _nolinebreak?>闲聊),将其变体称为AL–MG(增强型Lagrangian <?Pub / _nolinebreak?>多邻居闲聊)和AL–BG(增强型Lagrangian <?Pub / _nolinebreak?>广播八卦)。 AL–G算法基于增强的拉格朗日对偶函数。对偶变量通过标准的乘法器方法以较慢的时间尺度进行更新。为了更新原始变量,我们提出了一种新颖的,高斯-塞德尔型随机算法,可以在快速的时间尺度上进行。 AL–G仅在网络的直接邻居之间使用单向闲话通讯,并且可以抵抗随机链路故障。对于具有可靠通信(即无故障)的网络,简化的AL–BG(增强型Lagrangian <?Pub / _nolinebreak?>广播八卦)算法减少了通信,计算和数据存储成本。我们证明了所有提出的算法的收敛性,并通过仿真证明了两种应用的有效性: $ l_ {1} $ -正则化逻辑回归用于认知无线电网络的分类和协作频谱感知。

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