Reconstruction Limits for Repeated Differentially Private Aggregates: A Cramer-Rao Perspective on Query Geometry

Abstract

Repeated differentially private (DP) releases are often evaluated by transcript length or cumulative privacy accounting. We show that these quantities do not by themselves determine local reconstruction risk. For Gaussian-calibrated repeated statistical queries, the key object is the nuisance-profiled Fisher geometry of the release sequence: repetition helps only when new releases create identifiable directions after nuisance variables are removed. Thus, release geometry determines what can be locally identified, while the privacy accountant determines how precisely those directions can be estimated. We develop this principle in two settings. For labeled-target reconstruction with fixed-background IN/OUT averages, repeated copies collapse to a single target-versus-background contrast. The best linear unbiased estimator attains the Cramer-Rao bound, and additional copies provide only averaging gain; under Basic Composition this gain is dominated by the $\Theta(L^2\log L)$ noise penalty, whereas zCDP/RDP-style Gaussian accounting makes the risk order-flat. For static permutation-invariant releases, labels remain unidentified, but feature diversity can make the sorted participating multiset locally identifiable. For polynomial moments and smooth thresholds, the useful number of releases is governed by the balance between newly exposed eigendirections and accountant-induced noise growth. These results provide a local, mechanism-specific benchmark for value leakage in repeated private sensing and analytics.