Differential privacy (DP) is a leading framework for privacy protection, yet privacy mechanisms tailored to functional data analysis (FDA) remain relatively underexplored. For functional mean estimation, we investigate a fundamental question regarding the ordering of privatization and estimation by comparing two canonical pipelines: Fit-Then-Protect (FTP), which smooths trajectories before privatization, and Protect-Then-Fit (PTF), which privatizes observations at discrete time points before smoothing. When data are observed without error, we prove that FTP achieves strictly smaller mean squared error than PTF asymptotically (as the sample size increases) under the same $μ$-Gaussian differential privacy ($μ$-GDP) budget in both finite-dimensional Hilbert spaces and infinite-dimensional reproducing kernel Hilbert spaces (RKHS). Simulation studies support our theory and explore settings with observation error and fitting constraints, demonstrating that FTP outperforms PTF under small observation noise, whereas PTF can become preferable in high-noise settings. In addition to this comparison, we establish a novel GDP guarantee for Gaussian perturbations taking values in a Banach space, extending existing results beyond the RKHS setting.
Gaussian differential privacy (GDP) and zero-concentrated differential privacy (zCDP) are two competing privacy accounting frameworks, but they share similar minimax lower bounds in various estimation problems. Moreover, it is known that a mechanism satisfying $μ$-GDP satisfies $(μ^2/2)$-zCDP, whereas the converse fails to hold at the same calibration. In this paper, we establish the optimal reverse conversion, showing that every $ρ$-zCDP mechanism satisfies $μ_{\mathrm{zCDP}}(ρ)$-GDP, where $μ_{\mathrm{zCDP}}(ρ)=2Φ^{-1}((1+e^{-t_ρ})^{-1})$ and $t_ρ$ uniquely solves the equation $ρ=t_ρ\tanh(t_ρ/2)$. We also obtain simplified bounds $\sqrt{πρ}\leqμ_{\mathrm{zCDP}}(ρ)\leq\sqrt{8ρ}$, with sharp constants, which imply equivalence of the two frameworks up to constants. This explains why their minimax risks agree up to a constant rescaling of the privacy budget. We show that the squared conversion is strictly convex, which implies that converting zCDP to GDP before applying composition or group privacy yields a tighter GDP guarantee than converting afterward. We also derive the sharp conversion from bounded-range and logistic DP mechanisms (e.g., exponential mechanism) to GDP, where converting before composition but after group privacy gives the tighter guarantee.
Differential privacy is the leading mathematical framework for privacy protection, providing a probabilistic guarantee that safeguards individuals' private information when publishing statistics from a dataset. This guarantee is achieved by applying a randomized algorithm to the original data, which introduces unique challenges in data analysis by distorting inherent patterns. In particular, causal inference using observational data while preserving privacy is challenging because it requires a good covariate balance between treatment groups, but checking the covariate balance is challenging when privacy is a primary concern because releasing any statistic (e.g., t-statistic) for the balance check compromises privacy. Additionally, the performance of the privatized estimator critically depends on the choice of differential privacy mechanisms, and it remains unexplored how privacy-protecting causal inference could be conducted with observational data. In this article, we present a differentially private two-stage covariate balancing weighting estimator to infer causal effects from observational data. Our algorithm produces both point and interval estimators with statistical guarantees, such as covariate balance, rate optimality, and oracle budget allocation, under a given privacy budget.