Differentially Private Sliced Inverse Regression in the Federated Paradigm

Sliced inverse regression (SIR), which includes linear discriminant analysis (LDA) as a special case, is a popular and powerful dimension reduction tool. In this article, we extend SIR to address the challenges of decentralized data, prioritizing privacy and communication efficiency. Our approach, termed as federated sliced inverse regression (FSIR), facilitates distributed computing of the sufficient dimension reduction subspace among multiple clients, solely sharing local estimates to protect sensitive datasets from exposure. To guard against potential adversary attacks, FSIR employs diverse perturbation strategies, including a novel vectorized Gaussian mechanism that guarantees (ε,δ)-differential privacy at a low cost of statistical accuracy. Additionally, FSIR achieves a tight composition of various privacy mechanisms by adopting a hypothesis testing perspective on differential privacy. It also incorporates a collaborative feature screening procedure, enabling effective handling of high-dimensional client data with varying feature sets. Theoretical properties of FSIR are established for both low-dimensional and high-dimensional settings, supported by extensive numerical experiments and real data analysis. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.切片逆回归（Sliced inverse regression，SIR）是一类广受青睐且性能优异的降维工具，其特例涵盖线性判别分析（Linear Discriminant Analysis，LDA）。本文针对SIR展开推广研究，以解决分布式数据场景下的挑战，同时兼顾隐私保护需求与通信效率。我们所提出的方法被命名为联邦切片逆回归（Federated Sliced Inverse Regression，FSIR），该方法支持在多个客户端间完成充分降量子空间（sufficient dimension reduction subspace）的分布式计算，仅需共享局部估计结果，即可避免敏感数据集直接暴露。为抵御潜在的敌手攻击，FSIR采用了多样化的扰动策略，其中包括一种新型向量化高斯机制（vectorized Gaussian mechanism），该机制能够以极低的统计精度损耗实现(ε,δ)-差分隐私（(ε,δ)-differential privacy）保障。此外，FSIR借助差分隐私的假设检验视角，实现了各类隐私保护机制的紧密组合。该方法还集成了协同特征筛选（collaborative feature screening）流程，可有效处理特征集存在差异的高维客户端数据。本文针对低维和高维两种场景，分别确立了FSIR的理论性质，并通过大量数值实验与真实数据分析验证了其有效性。本文的补充材料可在线获取，其中包含了可复现本研究成果的相关材料的标准化说明。