Optimal Cox regression under federated differential privacy: coefficients and cumulative hazards

Abstract

We study two foundational problems in distributed survival analysis under federated differential privacy (FDP): estimation of the Cox regression coefficients and of the cumulative baseline hazard functions, allowing for heterogeneous per-sever sample sizes and privacy budgets. To quantify the fundamental cost of privacy, we derive minimax lower bounds together with upper bounds that match up to poly-logarithmic factors for the regression coefficients, thereby revealing server-level phase transitions between private and non-private regimes. We also consider a relaxed differential privacy framework with partially public information. Our analysis shows that the role of public covariates depends strongly on the privacy model. For cumulative hazard estimation, we propose a private tree-based version of the Breslow estimator for nonparametric integral estimation under FDP. As a by-product, this leads to a private survival function estimator that attains a nearly minimax optimal rate. Numerical experiments, including a real-data application, support the theoretical findings. The proposed methods are implemented in an accompanying R package FDPCox.