arXiv subjects

Jordan Awan

Publications and source records attributed to Jordan Awan.

At least 19 recordsLinked to original sources

arXiv2026

Optimal Conversion from Zero-Concentrated to Gaussian Differential Privacy

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.

cs.CR↗

arXiv2026

Differentially Private Covariate Balancing Causal Inference

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.

stat.ME↗

arXiv2026

Statistical Inference for Privatized Data with Unknown Sample Size

We develop both theory and algorithms to analyze privatized data in unbounded differential privacy (DP), where even the sample size is considered a sensitive quantity that requires privacy protection. We show that the distance between the sampling distributions under unbounded DP and bounded DP goes to zero as the sample size $n$ goes to infinity, provided that the noise used to privatize $n$ is at an appropriate rate; we also establish that Approximate Bayesian Computation (ABC)-type posterior distributions converge under similar assumptions. We further give asymptotic results in regimes where the privacy budgets vary, establishing similarity of sampling distributions as well as showing that the MLE in the unbounded setting converges to the bounded-DP MLE. To facilitate valid, finite-sample Bayesian inference on privatized data under unbounded DP, we propose a reversible jump MCMC algorithm which extends the data augmentation MCMC of Ju et al. (2022). We also propose a Monte Carlo EM algorithm to compute the MLE from privatized data in both bounded and unbounded DP. We apply our methodology to analyze a linear regression model as well as a 2019 American Time Use Survey Microdata File which we model using a Dirichlet distribution.

math.ST↗

arXiv2026

Particle Filter for Bayesian Inference on Privatized Data

Differential Privacy (DP) is a probabilistic framework that protects privacy while preserving data utility. To protect the privacy of the individuals in the dataset, DP requires adding a precise amount of noise to a statistic of interest; however, this noise addition alters the resulting sampling distribution, making statistical inference challenging. One of the main DP goals in Bayesian analysis is to make statistical inference based on the private posterior distribution. While existing methods have strengths in specific conditions, they can be limited by poor mixing, strict assumptions, or low acceptance rates. We propose a novel particle filtering algorithm, which features (i) consistent estimates, (ii) Monte Carlo error estimates and asymptotic confidence intervals, (iii) computational efficiency, and (iv) accommodation to a wide variety of priors, models, and privacy mechanisms with minimal assumptions. We empirically evaluate our algorithm through a variety of simulation settings as well as an application to a 2021 Canadian census dataset, demonstrating the efficacy and adaptability of the proposed sampler.

stat.CO↗

arXiv2026

Incomplete U-Statistics of Equireplicate Designs: Berry-Esseen Bound and Efficient Construction

U-statistics are a fundamental class of estimators that generalize the sample mean and underpin much of nonparametric statistics. Although extensively studied in both statistics and probability, key challenges remain: their high computational cost - addressed partly through incomplete U-statistics - and their non-standard asymptotic behavior in the degenerate case, which typically requires resampling methods for hypothesis testing. This paper presents a novel perspective on U-statistics, grounded in hypergraph theory and combinatorial designs. Our approach bypasses the traditional Hoeffding decomposition, the main analytical tool in this literature but one that is highly sensitive to degeneracy. By characterizing the dependence structure of a U-statistic, we derive a Berry-Esseen bound valid for incomplete U-statistics of deterministic designs, yielding conditions under which Gaussian limiting distributions can be established even in degenerate cases and when the order diverges. We also introduce efficient algorithms to construct incomplete U-statistics based on equireplicate designs, a subclass of deterministic designs that, in certain cases, achieve minimum variance. Beyond its theoretical contributions, our framework provides a systematic way to construct permutation-free counterparts to tests based on degenerate U-statistics, as demonstrated in experiments with kernel-based tests using the Maximum Mean Discrepancy and the Hilbert-Schmidt Independence Criterion.

math.ST↗

arXiv2026

Large-Sample Bayesian Approximations for Privatized Data

The increased use of differential privacy (DP) has allowed the sharing of large amounts of data while reducing the risk of disclosure of sensitive information at the individual level. However, the noise introduced by DP methods makes performing statistical inference more challenging. While various methods have been proposed to address different inferential tasks, they often require strong parametric assumptions and/or do not scale well with sample sizes (e.g. U.S. Census products). In response to these limitations, we propose an approximate Bayesian method to analyze privatized data products, which uses a two-step approach of imputing the confidential data and then sampling from the non-private posterior, and which is inspired by the method of Guha and Reiter (2025). We prove that this approximate sampler is asymptotically valid under mild assumptions. While this approach is motivated by Bayesian theory, we show through simulations that it provides conservative frequentist properties as well. We demonstrate the utility of our method by applying it in simulated settings as well as for an analysis on the drivers of homeownership via the 2022 American Community Survey.

stat.ME↗

arXiv2026

Optimal Debiased Inference on Privatized Data via Indirect Estimation and Parametric Bootstrap

We design a debiased parametric bootstrap framework for statistical inference from differentially private data. Existing usage of the parametric bootstrap on privatized data ignored or avoided handling possible biases introduced by the privacy mechanism, such as by clamping, a technique employed by the majority of privacy mechanisms. Ignoring these biases leads to under-coverage of confidence intervals and miscalibrated type I errors of hypothesis tests, due to the inconsistency of parameter estimates based on the privatized data. We propose using the indirect inference method to estimate the parameter values consistently, and we use the improved estimator in parametric bootstrap for inference. To implement the indirect estimator, we present a novel simulation-based, adaptive approach along with the theory that establishes the consistency of the corresponding parametric bootstrap estimates, confidence intervals, and hypothesis tests. In particular, we prove that our adaptive indirect estimator achieves the minimum asymptotic variance among all ``well-behaved'' consistent estimators based on the released summary statistic. Our simulation studies show that our framework produces confidence intervals with well-calibrated coverage and performs hypothesis testing with the correct type I error, giving state-of-the-art performance for inference in several settings.

stat.ME↗

arXiv2026

Public Good Provision under Locally Private Signals

We study public-good provision when a planner observes agents' preferences only through a fixed local-privacy channel that randomizes each report before it reaches the planner. We characterize the optimal reduced-form allocation: the project is implemented when an aggregate posterior score is positive, where each agent's score combines the posterior expected valuation and posterior virtual value. Privacy enters through these posterior objects, muting the responsiveness of provision to private preferences and, under weak monotone likelihood ratios, potentially generating pooling. We then distinguish the optimal reduced-form allocation from its implementation through signal-measurable transfers: the required transfers solve a Fredholm integral equation whose solution is unique under completeness when it exists, while existence requires a separate range condition. Maximum reduced-form revenue exhibits three population regimes: it is asymptotically linear, of square-root order, or exponentially small according as the lower endpoint of the valuation distribution is positive, zero, or negative. Finally, welfare comparisons depend on the privacy calibration. At a common noise scale, Laplace Blackwell-dominates logistic noise, while under a common tight $μ$-GDP calibration the ordering reverses for the maximally separated binary endpoint experiment. Thus the preferred privacy channel depends on the standard used to hold privacy fixed.

econ.TH↗

arXiv2026

Near-Optimal Private Tests for Simple and MLR Hypotheses

We develop a near-optimal testing procedure under the framework of Gaussian differential privacy for simple as well as one- and two-sided tests under monotone likelihood ratio conditions. Our mechanism is based on a private mean estimator with data-driven clamping bounds, whose population risk matches the private minimax rate up to logarithmic factors. Using this estimator, we construct private test statistics that achieve the same asymptotic relative efficiency as the non-private, most powerful tests while maintaining conservative type I error control. In addition to our theoretical results, our numerical experiments show that our private tests outperform competing DP methods and offer comparable power to the non-private most powerful tests, even at moderately small sample sizes and privacy loss budgets.

stat.ML↗

arXiv2026

Beyond Data Splitting: Full-Data Conformal Prediction by Differential Privacy

Privacy protection and uncertainty quantification are increasingly important in data-driven decision making. Conformal prediction provides finite-sample marginal coverage, but existing private approaches often rely on data splitting, reducing the effective sample size. We propose a full-data privacy-preserving conformal prediction framework that avoids splitting. Our framework leverages stability induced by differential privacy to control the gap between in-sample and out-of-sample conformal scores, and pairs this with a conservative private quantile routine designed to prevent under-coverage. We show that a generic differential privacy guarantee yields a universal coverage floor, yet cannot generally recover the nominal $1-α$ level. We then provide a refined, mechanism-specific stability analysis and yields asymptotic recovery of the nominal level. Experiments demonstrate sharper prediction sets than the split-based private baseline.

stat.ML↗

arXiv2025

Differentially Private Bootstrap: New Privacy Analysis and Inference Strategies

Differentially private (DP) mechanisms protect individual-level information by introducing randomness into the statistical analysis procedure. Despite the availability of numerous DP tools, there remains a lack of general techniques for conducting statistical inference under DP. We examine a DP bootstrap procedure that releases multiple private bootstrap estimates to infer the sampling distribution and construct confidence intervals (CIs). Our privacy analysis presents new results on the privacy cost of a single DP bootstrap estimate, applicable to any DP mechanism, and identifies some misapplications of the bootstrap in the existing literature. For the composition of the DP bootstrap, we present a numerical method to compute the exact privacy cost of releasing multiple DP bootstrap estimates, and using the Gaussian-DP (GDP) framework (Dong et al., 2022), we show that the release of $B$ DP bootstrap estimates from mechanisms satisfying $(μ/\sqrt{(2-2/\mathrm{e})B})$-GDP asymptotically satisfies $μ$-GDP as $B$ goes to infinity. Then, we perform private statistical inference by post-processing the DP bootstrap estimates. We prove that our point estimates are consistent, our standard CIs are asymptotically valid, and both enjoy optimal convergence rates. To further improve the finite performance, we use deconvolution with DP bootstrap estimates to accurately infer the sampling distribution. We derive CIs for tasks such as population mean estimation, logistic regression, and quantile regression, and we compare them to existing methods using simulations and real-world experiments on 2016 Canada Census data. Our private CIs achieve the nominal coverage level and offer the first approach to private inference for quantile regression.

stat.ML↗

arXiv2025

Best Linear Unbiased Estimate from Privatized Contingency Tables

In differential privacy (DP) mechanisms, it can be beneficial to release "redundant" outputs, where some quantities can be estimated in multiple ways by combining different privatized values. Indeed, the DP 2020 Decennial Census products published by the U.S. Census Bureau consist of such redundant noisy counts. When redundancy is present, the DP output can be improved by enforcing self-consistency (i.e., estimators obtained using different noisy counts result in the same value), and we show that the minimum variance processing is a linear projection. However, standard projection algorithms require excessive computation and memory, making them impractical for large-scale applications such as the Decennial Census. We propose the Scalable Efficient Algorithm for Best Linear Unbiased Estimate (SEA BLUE), based on a two-step process of aggregation and differencing that 1) enforces self-consistency through a linear and unbiased procedure, 2) is computationally and memory efficient, 3) achieves the minimum variance solution under certain structural assumptions, and 4) is empirically shown to be robust to violations of these structural assumptions. We propose three methods of calculating confidence intervals from our estimates, under various assumptions. Finally, we apply SEA BLUE to two 2010 Census demonstration products, illustrating its scalability and validity.

stat.CO↗

arXiv2024

Optimizing Noise for $f$-Differential Privacy via Anti-Concentration and Stochastic Dominance

In this paper, we establish anti-concentration inequalities for additive noise mechanisms which achieve $f$-differential privacy ($f$-DP), a notion of privacy phrased in terms of a tradeoff function $f$ which limits the ability of an adversary to determine which individuals were in the database. We show that canonical noise distributions (CNDs), proposed by Awan and Vadhan (2023), match the anti-concentration bounds at half-integer values, indicating that their tail behavior is near-optimal. We also show that all CNDs are sub-exponential, regardless of the $f$-DP guarantee. In the case of log-concave CNDs, we show that they are the stochastically smallest noise compared to any other noise distributions with the same privacy guarantee. In terms of integer-valued noise, we propose a new notion of discrete CND and prove that a discrete CND always exists, can be constructed by rounding a continuous CND, and that the discrete CND is unique when designed for a statistic with sensitivity 1. We further show that the discrete CND at sensitivity 1 is stochastically smallest compared to other integer-valued noises. Our theoretical results shed light on the different types of privacy guarantees possible in the $f$-DP framework and can be incorporated in more complex mechanisms to optimize performance.

cs.CR↗

arXiv2024

Simulation-based, Finite-sample Inference for Privatized Data

Privacy protection methods, such as differentially private mechanisms, introduce noise into resulting statistics which often produces complex and intractable sampling distributions. In this paper, we propose a simulation-based "repro sample" approach to produce statistically valid confidence intervals and hypothesis tests, which builds on the work of Xie and Wang (2022). We show that this methodology is applicable to a wide variety of private inference problems, appropriately accounts for biases introduced by privacy mechanisms (such as by clamping), and improves over other state-of-the-art inference methods such as the parametric bootstrap in terms of the coverage and type I error of the private inference. We also develop significant improvements and extensions for the repro sample methodology for general models (not necessarily related to privacy), including 1) modifying the procedure to ensure guaranteed coverage and type I errors, even accounting for Monte Carlo error, and 2) proposing efficient numerical algorithms to implement the confidence intervals and $p$-values.

math.ST↗

arXiv2024

Structure and Sensitivity in Differential Privacy: Comparing K-Norm Mechanisms

Differential privacy (DP), provides a framework for provable privacy protection against arbitrary adversaries, while allowing the release of summary statistics and synthetic data. We address the problem of releasing a noisy real-valued statistic vector $T$, a function of sensitive data under DP, via the class of $K$-norm mechanisms with the goal of minimizing the noise added to achieve privacy. First, we introduce the sensitivity space of $T$, which extends the concepts of sensitivity polytope and sensitivity hull to the setting of arbitrary statistics $T$. We then propose a framework consisting of three methods for comparing the $K$-norm mechanisms: 1) a multivariate extension of stochastic dominance, 2) the entropy of the mechanism, and 3) the conditional variance given a direction, to identify the optimal $K$-norm mechanism. In all of these criteria, the optimal $K$-norm mechanism is generated by the convex hull of the sensitivity space. Using our methodology, we extend the objective perturbation and functional mechanisms and apply these tools to logistic and linear regression, allowing for private releases of statistical results. Via simulations and an application to a housing price dataset, we demonstrate that our proposed methodology offers a substantial improvement in utility for the same level of risk.

stat.ME↗

arXiv2024

Formal Privacy Guarantees with Invariant Statistics

Motivated by the 2020 US Census products, this paper extends differential privacy (DP) to address the joint release of DP outputs and nonprivate statistics, referred to as invariant. Our framework, Semi-DP, redefines adjacency by focusing on datasets that conform to the given invariant, ensuring indistinguishability between adjacent datasets within invariant-conforming datasets. We further develop customized mechanisms that satisfy Semi-DP, including the Gaussian mechanism and the optimal $K$-norm mechanism for rank-deficient sensitivity spaces. Our framework is applied to contingency table analysis which is relevant to the 2020 US Census, illustrating how Semi-DP enables the release of private outputs given the one-way margins as the invariant. Additionally, we provide a privacy analysis of the 2020 US Decennial Census using the Semi-DP framework, revealing that the effective privacy guarantees are weaker than advertised.

cs.CR↗

arXiv2024

One Step to Efficient Synthetic Data

A common approach to synthetic data is to sample from a fitted model. We show that under general assumptions, this approach results in a sample with inefficient estimators and whose joint distribution is inconsistent with the true distribution. Motivated by this, we propose a general method of producing synthetic data, which is widely applicable for parametric models, has asymptotically efficient summary statistics, and is both easily implemented and highly computationally efficient. Our approach allows for the construction of both partially synthetic datasets, which preserve certain summary statistics, as well as fully synthetic data which satisfy the strong guarantee of differential privacy (DP), both with the same asymptotic guarantees. We also provide theoretical and empirical evidence that the distribution from our procedure converges to the true distribution. Besides our focus on synthetic data, our procedure can also be used to perform approximate hypothesis tests in the presence of intractable likelihood functions.

math.ST↗

arXiv2023

Differentially Private Topological Data Analysis

This paper is the first to attempt differentially private (DP) topological data analysis (TDA), producing near-optimal private persistence diagrams. We analyze the sensitivity of persistence diagrams in terms of the bottleneck distance, and we show that the commonly used Čech complex has sensitivity that does not decrease as the sample size $n$ increases. This makes it challenging for the persistence diagrams of Čech complexes to be privatized. As an alternative, we show that the persistence diagram obtained by the $L^1$-distance to measure (DTM) has sensitivity $O(1/n)$. Based on the sensitivity analysis, we propose using the exponential mechanism whose utility function is defined in terms of the bottleneck distance of the $L^1$-DTM persistence diagrams. We also derive upper and lower bounds of the accuracy of our privacy mechanism; the obtained bounds indicate that the privacy error of our mechanism is near-optimal. We demonstrate the performance of our privatized persistence diagrams through simulations as well as on a real dataset tracking human movement.

stat.ML↗