分布式精度矩阵整合估计的最优整合估计与异质性调整
Optimal Integrative Estimation for Distributed Precision Matrices with Heterogeneity Adjustment
AI总结：
本文提出HEAT和IteHEAT方法，用于高效估计分布式异质精度矩阵，通过优化通信和计算效率，实现统计最优性。
摘要：
分布式学习为多源数据集的整合分析提供了一个实用的解决方案，特别是在隐私或通信限制下。然而，解决前景分布异质性和确保通信效率在分布式统计分析中提出了重大挑战。在本文中，我们专注于分布式异质精度矩阵的整合估计，这与联合精度矩阵估计密切相关，其中计算高效的算法和统计最优性理论仍不完善。为了解决这些挑战，我们引入了一种新的HEterogeneity-adjusted Aggregating and Thresholding（HEAT）方法用于分布式整合估计。HEAT被设计为既通信又计算高效，并通过在各种整合损失下建立收敛速率和相应的minimax下界来证明其统计最优性。为了增强HEAT的最优性，我们进一步提出了一种迭代HEAT（IteHEAT）方法。通过通过多轮通信迭代改进HEAT估计器的高阶误差，IteHEAT实现了几何收缩率的收敛速度。广泛的模拟和真实数据应用验证了HEAT和IteHEAT方法的数值性能。
英文摘要：
Distributed learning offers a practical solution for the integrative analysis of multi-source datasets, especially under privacy or communication constraints. However, addressing prospective distributional heterogeneity and ensuring communication efficiency pose significant challenges on distributed statistical analysis. In this article, we focus on integrative estimation of distributed heterogeneous precision matrices, a crucial task related to joint precision matrix estimation where computation-efficient algorithms and statistical optimality theories are still underdeveloped. To tackle these challenges, we introduce a novel HEterogeneity-adjusted Aggregating and Thresholding (HEAT) approach for distributed integrative estimation. HEAT is designed to be both communication- and computation-efficient, and we demonstrate its statistical optimality by establishing the convergence rates and the corresponding minimax lower bounds under various integrative losses. To enhance the optimality of HEAT, we further propose an iterative HEAT (IteHEAT) approach. By iteratively refining the higher-order errors of HEAT estimators through multi-round communications, IteHEAT achieves geometric contraction rates of convergence. Extensive simulations and real data applications validate the numerical performance of HEAT and IteHEAT methods.
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