Exact Asymptotic Efficiency under zCDP: Diameter-Constrained Information Geometry

RESUMO

Protecting individual privacy has become a central and urgent concern in modern data analysis, given the vast quantities of data now generated and processed. In this paper, we develop a systematic theory of exact asymptotic efficiency for regular parametric estimation under central zero-concentrated differential privacy. In the privacy regime, the governing object is a diameter-constrained information region: the set of information matrices generated by statistics with diameter at most one. For weighted quadratic loss, we show that the exact local minimax risk is an inverse information variational functional over this region. More generally, a mixed information region yields a unified efficiency constant across different regimes, covering the privacy regime and classical Fisher efficiency. A matching estimator releases the empirical mean of a nearly optimal bounded statistic with Gaussian noise and locally inverts its population moment map. Our theory differs from classical efficiency theory in its set-valued information geometry and loss-dependent efficient estimator. As examples, we provide closed-form constants and optimal procedures for various concrete models, including one-dimensional regular families, Gaussian means, categorical probability vectors, and regression models among others. The theory also transfers to Gaussian differential privacy through an exact parameter rescaling.