有向网络上的分布式推理：性能极限和最优设计
Distributed inference over directed networks: Performance limits and optimal design
摘要：
我们发现大偏差率基于共识的分布式推理有向网络。当拓扑结构是确定性的，我们建立大偏差原则，并找到相应的速率函数，在所有节点相等。我们表明，与网络相关联的随机权重矩阵的速率函数的依赖性完全捕获其左特征向量对应的单位特征值。此外，当传感器的观察是高斯的，速率函数承认一个封闭的形式表示。出于这些意见，我们制定了最佳的网络设计问题，找到左特征向量，达到最高值的速率函数，对于给定的目标精度。因此，该特征向量使推理算法达到期望精度所需的时间最小化。对于高斯观测，我们表明，网络设计问题可以制定为一个半定（凸）计划，因此可以有效地解决。当观察是相同的分布在代理，系统表现出一个有趣的属性：率函数的图形总是位于之间的图形的率函数的一个孤立的节点和率函数的融合中心，可以访问所有的意见。我们证明了这一基本属性，即使当拓扑结构和相关的系统矩阵随时间随机变化，任意分布。由于其假设的一般性，后者的结果需要比标准大偏差工具更微妙的技术，有助于大偏差的一般理论。
英文摘要：
We find large deviations rates for consensus-based distributed inference for directed networks. When the topology is deterministic, we establish the large deviations principle and find exactly the corresponding rate function, equal at all nodes. We show that the dependence of the rate function on the stochastic weight matrix associated with the network is fully captured by its left eigenvector corresponding to the unit eigenvalue. Further, when the sensors' observations are Gaussian, the rate function admits a closed form expression. Motivated by these observations, we formulate the optimal network design problem of finding the left eigenvector which achieves the highest value of the rate function, for a given target accuracy. This eigenvector therefore minimizes the time that the inference algorithm needs to reach the desired accuracy. For Gaussian observations, we show that the network design problem can be formulated as a semidefinite (convex) program, and hence can be solved efficiently. When observations are identically distributed across agents, the system exhibits an interesting property: the graph of the rate function always lies between the graphs of the rate function of an isolated node and the rate function of a fusion center that has access to all observations. We prove that this fundamental property holds even when the topology and the associated system matrices change randomly over time, with arbitrary distribution. Due to generality of its assumptions, the latter result requires more subtle techniques than the standard large deviations tools, contributing to the general theory of large deviations.
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