高维共识：大规模网络中的学习
Higher Dimensional Consensus: Learning in Large-Scale Networks
摘要：
提出了一种适用于大规模网络的高维一致性算法。HDC推广了众所周知的平均共识算法。它将大规模网络的节点分为锚节点和传感器节点。邻居是其状态在HDC迭代中固定的节点，而传感器是将其状态更新为相邻状态的线性组合的节点。在适当的条件下，我们证明了传感器状态收敛到锚状态的线性组合。通过锚的概念，HDC在一个统一的框架中捕获了几个有趣的网络任务，包括分布式传感器定位，领导者-追随者，分布式Jacobi求解线性代数方程组，当然还有平均共识。在许多网络应用中，学习分布式线性算法的权值以使传感器收敛到期望的状态是很有意义的。我们把这个逆问题称为HDC学习问题。我们将HDC中的学习作为一个约束非凸优化问题，我们将其置于多目标优化（MOP）的框架中，并将帕累托最优性应用于其中。我们证明了MOP解决方案和帕累托前，我们从中获得的解决方案，在HDC学习分析相关的属性。最后，本文展示了MOP方法如何解决资源受限网络中HDC学习中出现的有趣权衡（收敛速度与最终状态质量）。
英文摘要：
The paper presents higher dimension consensus (HDC) for large-scale networks. HDC generalizes the well-known average-consensus algorithm. It divides the nodes of the large-scale network into anchors and sensors. Anchors are nodes whose states are fixed over the HDC iterations, whereas sensors are nodes that update their states as a linear combination of the neighboring states. Under appropriate conditions, we show that the sensor states converge to a linear combination of the anchor states. Through the concept of anchors, HDC captures in a unified framework several interesting network tasks, including distributed sensor localization, leader-follower, distributed Jacobi to solve linear systems of algebraic equations, and, of course, average-consensus. In many network applications, it is of interest to learn the weights of the distributed linear algorithm so that the sensors converge to a desired state. We term this inverse problem the HDC learning problem. We pose learning in HDC as a constrained non-convex optimization problem, which we cast in the framework of multi-objective optimization (MOP) and to which we apply Pareto optimality. We prove analytically relevant properties of the MOP solutions and of the Pareto front from which we derive the solution to learning in HDC. Finally, the paper shows how the MOP approach resolves interesting tradeoffs (speed of convergence versus quality of the final state) arising in learning in HDC in resource constrained networks.
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