Decentralized Learning of Tree-Structured Gaussian Graphical Models from Noisy Data

Abstract

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.

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References in corpus (9)

• Speeding Up Distributed Machine Learning Using Codes
• Robust Graph Learning from Noisy Data
• A Large-Deviation Analysis of the Maximum-Likelihood Learning of Markov Tree Structures
• Learning Gaussian Tree Models: Analysis of Error Exponents and Extremal Structures
• Marginal Likelihoods for Distributed Parameter Estimation of Gaussian Graphical Models
• Distributed Learning over Unreliable Networks
• Robust Estimation of Tree Structured Ising Models
• Learning of Tree-Structured Gaussian Graphical Models on Distributed Data under Communication Constraints
• Optimal Rates for Learning Hidden Tree Structures