This paper studies the decentralized learning of tree-structured Gaussian graphical models (GGMs) from noisy data. In decentralized learning, data set is distributed across different machines (sensors), and GGMs are widely used to model complex networks such as gene regulatory networks and social networks. The proposed decentralized learning uses the Chow-Liu algorithm for estimating the tree-structured GGM. In previous works, upper bounds on the probability of incorrect tree structure recovery were given mostly without any practical noise for simplification. While this paper investigates the effects of three common types of noisy channels: Gaussian, Erasure, and binary symmetric channel. For Gaussian channel case, to satisfy the failure probability upper bound $\delta > 0$ in recovering a $d$-node tree structure, our proposed theorem requires only $\mathcal{O}(\log(\frac{d}{\delta}))$ samples for the smallest sample size ($n$) comparing to the previous literature \cite{Nikolakakis} with $\mathcal{O}(\log^4(\frac{d}{\delta}))$ samples by using the positive correlation coefficient assumption that is used in some important works in the literature. Moreover, the approximately bounded Gaussian random variable assumption does not appear in \cite{Nikolakakis}. Given some knowledge about the tree structure, the proposed Algorithmic Bound will achieve obviously better performance with small sample size (e.g., $< 2000$) comparing with formulaic bounds. Finally, we validate our theoretical results by performing simulations on synthetic data sets.
翻译:本文研究了从杂乱的数据中分散学习树结构高斯的图形模型( GGMs) 。 在分散学习中, 数据集分布在不同机器( 传感器), GGMs 被广泛用于模拟基因监管网络和社会网络等复杂网络。 拟议的分散学习使用 Chow- Liu 算法来估计树结构GGM 。 在先前的作品中, 给出不正确的树结构恢复概率的上限大多没有任何实际的简化噪音 。 虽然本文调查三种常见的噪音频道类型: Gausian、 Erasure 和二进制对称频道的效应。 对于 Gaussian 频道案例, 为了在恢复$d$- node 树结构时满足失败概率上限$\delta > 0美元。 我们提议的语标仅需要 $mathcal{O} (\log (\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\