带噪声链路的网络约束平均最优算法的非渐近分析
Non-Asymptotic Analysis of an Optimal Algorithm for Network-Constrained Averaging with Noisy Links
摘要：
网络约束平均的问题是使用只能沿着图的边缘传递消息的算法来计算分布在图G中的一组值的平均值。我们研究这个问题在嘈杂的设置，其中沿每个链路的通信是由加性高斯白噪声信道建模。我们提出了一个两阶段的分散算法，我们使用随机逼近方法与谱图理论提供具体的（非渐近）的均方误差的界限。找到了这样的界限后，我们分析了实现均方误差δ所需的迭代次数T_G（n; δ）如何作为图拓扑和节点数n的函数。以前的工作提供了保证与第二个最小的特征值的拉普拉斯算子的迭代次数成反比。本文给出了一个算法，减少这种图的依赖，图直径，这是最好的缩放可能。
英文摘要：
The problem of network-constrained averaging is to compute the average of a set of values distributed throughout a graph G using an algorithm that can pass messages only along graph edges. We study this problem in the noisy setting, in which the communication along each link is modeled by an additive white Gaussian noise channel. We propose a two-phase decentralized algorithm, and we use stochastic approximation methods in conjunction with the spectral graph theory to provide concrete (non-asymptotic) bounds on the mean-squared error. Having found such bounds, we analyze how the number of iterations T_G(n; δ) required to achieve mean-squared error δ scales as a function of the graph topology and the number of nodes n. Previous work provided guarantees with the number of iterations scaling inversely with the second smallest eigenvalue of the Laplacian. This paper gives an algorithm that reduces this graph dependence to the graph diameter, which is the best scaling possible.
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