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.