SIAM Journal on Optimization, Volume 30, Issue 2, Page 1191-1222, January 2020.
In this paper, we study network linear equations subject to digital communications with a finite data rate, where each node is associated with one equation from a system of linear equations. Each node holds a dynamic state and interacts with its neighbors through an undirected connected graph, where along each link the pair of nodes share information. Due to the data rate constraint, each node builds an encoder/decoder pair, with which it produces transmitted messages with a zooming-in finite-level uniform quantizer and also generates estimates of its neighbors' states from the received signals. We then propose a distributed quantized algorithm and show that when the network linear equations admit a unique solution, each node's state is driven to that solution exponentially fast. We further analyze the asymptotic rate of convergence and show that a larger number of quantization levels leads to a faster convergence rate although the rate is still fundamentally bounded by the inherent network structure and the linear equations. In addition, we establish a bound on the total number of communication bits required to obtain a solution with a prescribed accuracy. When a unique least-squares solution exists, we show that the algorithm can compute such a solution with a suitably selected time-varying step-size inherited from the encoder and zooming-in quantizer dynamics. In both cases, a minimal data rate is shown to be enough for guaranteeing the desired convergence when the algorithm parameters are properly chosen. These results ensure the applicability of various network linear equation solvers when peer-to-peer communication is digital.

中文翻译：

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具有有限数据速率的线性算法的分布式算法

SIAM优化杂志，第30卷，第2期，第1191-1222页，2020年1月。
在本文中，我们研究了以有限数据速率进行数字通信的网络线性方程，其中每个节点都与线性方程组中的一个方程相关联。每个节点都具有动态状态，并通过无向连接图与其邻居互动，其中沿每条链接，一对节点共享信息。由于数据速率的限制，每个节点都构建了一个编码器/解码器对，它使用放大有限级均匀量化器生成一对已发送消息，并从接收到的信号中生成其邻居状态的估计值。然后，我们提出了一种分布式量化算法，并证明了当网络线性方程式接受唯一解时，每个节点的状态都以指数级的速度被驱动到该解。我们进一步分析了收敛的渐近速率，并表明，尽管量化速率仍从本质上受固有的网络结构和线性方程式限制，但大量的量化级别会导致更快的收敛速率。此外，我们为获得具有规定精度的解决方案所需的通信位数的总数设置了界限。当一个独特的最小二乘解存在，我们表明，该算法可以计算与从编码器和缩放-在量化动力学继承适当选择的时间变化的步长这样的解决方案。在这两种情况下，当正确选择算法参数时，最小数据速率都足以保证所需的收敛性。