We establish sharp minimax limits for two-sample testing of H\"older-smooth densities under central differential privacy. Given two independent samples, the goal is to decide whether the underlying distributions are identical or separated in $L_1$ distance, while releasing only an $\varepsilon$-differentially private decision. We show that privacy changes the classical smooth-testing boundary through multiple regimes: the optimal separation radius is the maximum of four terms, consisting of the classical nonprivate rate and three distinct privacy-induced barriers. Which barrier is active depends on the privacy budget and the smoothness-to-dimension ratio, yielding a sharp phase diagram. Our upper bound discretizes the samples, applies a private discrete two-sample test to the resulting histograms, and chooses the bin resolution to balance approximation bias, sampling fluctuations, and privacy noise. The procedure also admits a permutation-calibrated implementation with finite-sample type~I error control. For the lower bounds, we combine smooth perturbation constructions with privacy-specific coupling and transport inequalities, showing that all four terms are unavoidable. Finally, when the smoothness is unknown, we develop a multiscale private test that attains the optimal adaptive rate and prove a matching lower bound. Adaptation costs exactly an iterated-logarithmic factor, and this cost appears only in the classical nonprivate term.
Local differential privacy (LDP) protects individuals in a dataset by perturbing each measurement before release. For real-valued data, a widely used mechanism is additive Laplace noise. We study the problem of estimating the distribution of the latent confidential data from the privatized observations via the nonparametric maximum likelihood estimator (NPMLE) under an i.i.d. sampling model. We first show that under the Laplace convolution model, the NPMLE admits a finite-dimensional reformulation in which the support is restricted to the observation set. This reduces the original infinite-dimensional optimization over all mixing distributions to an $n$-dimensional convex optimization over mixture weights, where $n$ is the sample size. We then study the statistical convergence of the NPMLE under the 1-Wasserstein distance, and explicitly connect its convergence rate with the privacy noise scale. Allowing the privacy noise level to change with the sample size, our analysis shows that the NPMLE remains consistent when the Laplace noise grows at a rate slower than $n^{3/16}$. Conversely, when the Laplace noise is of the order $\sqrt n$ or larger, no estimator can achieve uniformly consistent recovery of the latent distribution.
Yifei Xiong, Nianqiao Ju, Vinayak A. Rao· 0 citations
We study the fundamental problem of estimating the dependence structure of a centered stationary Gaussian process under local differential privacy (LDP). In this setting, the spectral density characterizes the dependence structure of the data and is the quantity to be estimated. Our main contribution is to close the open $\alpha^2$-versus-$\alpha^4$ gap between the previously known lower and upper bounds on the minimax rate. Specifically, we establish a minimax lower bound showing that, over Sobolev-type classes of spectral densities, the effective sample size in the high-privacy regime is $N\alpha^4$, rather than the usual $N\alpha^2$ arising for independent observations. This additional privacy cost is caused by the temporal dependence between the observations rather than by their marginal distributions. The proof relies on a contraction bound for privatized dependent Gaussian observations. Our second contribution is a matching upper bound, free of the polylogarithmic losses present in previous work. Rather than applying a generic privatization scheme to classical estimators, we construct a problem-specific procedure attaining the rate identified by our lower bound. Beyond closing the gaps in spectral density estimation, we apply the tools developed for this problem to several related questions. We (i) close the logarithmic gap for fixed-lag autocovariance estimation, (ii) show that the $\alpha^4$ cost arises locally around every spectral density bounded away from zero, and (iii) establish that classical asymptotic equivalence with an independent Gaussian experiment generally fails under LDP.
The results show that sharp converses for minimax quantiles require adapting the information measure to the recovery resolution, whether exact or approximate, and to the tail behaviour of the likelihood ratio.
Differential privacy has become the standard for generating privacy-protected data releases. However, differential privacy does not translate intuitively to disclosure risk. In particular, it remains unclear how much an adversary's belief about an individual's inclusion in a dataset can change after observing a protected release. To address this question, we derive upper and lower bounds on the posterior-to-prior ratios of inclusion beliefs under bounded probabilistic and approximate differential privacy. By assuming a worst-case adversary with all-but-one auxiliary information, i.e., knowledge of all except for one of the participants in a dataset, we obtain bounds that apply to any adversary. Because these bounds may fail with non-zero probability, we study the corresponding failure probability for the Gaussian mechanism. We derive a theoretical upper limit on this probability and compare it with Monte Carlo estimates across a wide range of parameter settings. The observed failure rate is several orders of magnitude smaller than its theoretical upper limit, indicating that the latter is highly conservative. These findings suggest that the inferential privacy guarantees provided by differentially private mechanisms may be substantially stronger in practice than what is implied by the theoretical upper limit.
Jan Reiter Sørensen, H. S. Christensen, Rasmus Rask Kragh Jørgensen et al.· 0 citations
A comparative study of four audits applicable to pre-trained, black-box face generators, which consistently reveal substantial identity distinguishability while reporting markedly different epsilon estimates that reflect each method's distinct assumptions and finite-sample treatment.
Arman Zareian Jahromi, Vishnu Bondalakunta, Mohammad Akbar Bin Shah et al.· 0 citations
Because 2SLS is built from sample averages, a small number of observations can have a disproportionate effect on estimates and inference. We introduce W-2SLS, a simple drop-in robustification that replaces these averages by quantile-winsorized means. We analyze W-2SLS under adversarial contamination, which permits both the identities and the reported values of the contaminated observations to depend on the realized clean sample and therefore accommodates targeted or strategic manipulation. Under finite $m$-th moments, W-2SLS attains the minimax-sharp rate $\eta_{n}^{1-\frac1m}+n^{-1/2}$, where $\eta_n$ is the fraction of observations that may be altered. Matching lower bounds identify the exact contamination thresholds for uniform consistency, root-$n$ estimation, and centered Gaussian inference with the same first-order law as clean-sample 2SLS. When $\sqrt{n}\eta_{n}^{1-\frac1m}\to 0$ robustness is first-order free. We also construct feasible heteroskedasticity-robust inference and a winsorized Anderson--Rubin test valid under weak identification and adversarial contamination. Finally, even without contamination, ordinary 2SLS can have poor uniform finite-sample concentration, whereas W-2SLS admits confidence-calibrated sub-Gaussian deviation guarantees.
A. B. Kock, David Preinerstorfer· 0 citations
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