# About the Cost of Central Privacy in Density Estimation

arXiv: https://arxiv.org/abs/2306.14535 · HTML: https://mlsanity.com/paper/2306.14535/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 66 references that are in the map.
- First posted on arXiv 26 Jun 2023.

## Builds on (most cited first, 28 of 28)

- [Deep Learning with Differential Privacy](https://mlsanity.com/paper/1607.00133/) (2016)
- [Renyi Differential Privacy](https://mlsanity.com/paper/1702.07476/) (2017)
- [Concentrated Differential Privacy: Simplifications, Extensions, and Lower Bounds](https://mlsanity.com/paper/1605.02065/) (2016)
- [RAPPOR: Randomized Aggregatable Privacy-Preserving Ordinal Response](https://mlsanity.com/paper/1407.6981/) (2014)
- [Collecting Telemetry Data Privately](https://mlsanity.com/paper/1712.01524/) (2017)
- [Concentrated Differential Privacy](https://mlsanity.com/paper/1603.01887/) (2016)
- [Minimax Optimal Procedures for Locally Private Estimation](https://mlsanity.com/paper/1604.02390/) (2016)
- [Finite Sample Differentially Private Confidence Intervals](https://mlsanity.com/paper/1711.03908/) (2017)
- [Privately Learning High-Dimensional Distributions](https://mlsanity.com/paper/1805.00216/) (2018)
- [CoinPress: Practical Private Mean and Covariance Estimation](https://mlsanity.com/paper/2006.06618/) (2020)
- [Differentially Private Assouad, Fano, and Le Cam](https://mlsanity.com/paper/2004.06830/) (2020)
- [Improved Rates for Differentially Private Stochastic Convex Optimization with Heavy-Tailed Data](https://mlsanity.com/paper/2106.01336/) (2021)
- [From Robustness to Privacy and Back](https://mlsanity.com/paper/2302.01855/) (2023)
- [Local Privacy, Data Processing Inequalities, and Statistical Minimax Rates](https://mlsanity.com/paper/1302.3203/) (2013)
- [Differentially Private Testing of Identity and Closeness of Discrete Distributions](https://mlsanity.com/paper/1707.05128/) (2017)
- [Near Instance-Optimality in Differential Privacy](https://mlsanity.com/paper/2005.10630/) (2020)
- [Unified lower bounds for interactive high-dimensional estimation under information constraints](https://mlsanity.com/paper/2010.06562/) (2020)
- [Local differential privacy: Elbow effect in optimal density estimation and adaptation over Besov ellipsoids](https://mlsanity.com/paper/1903.01927/) (2019)
- [Inference under Information Constraints III: Local Privacy Constraints](https://mlsanity.com/paper/2101.07981/) (2021)
- [Optimal Rates for Nonparametric Density Estimation under Communication Constraints](https://mlsanity.com/paper/2107.10078/) (2021)
- [Strongly universally consistent nonparametric regression and classification with privatised data](https://mlsanity.com/paper/2011.00216/) (2020)
- [Information-constrained optimization: can adaptive processing of gradients help?](https://mlsanity.com/paper/2104.00979/) (2021)
- [On the Statistical Complexity of Estimation and Testing under Privacy Constraints](https://mlsanity.com/paper/2210.02215/) (2022)
- [Private Quantiles Estimation in the Presence of Atoms](https://mlsanity.com/paper/2202.08969/) (2022)
- [A Polynomial Time, Pure Differentially Private Estimator for Binary Product Distributions](https://mlsanity.com/paper/2304.06787/) (2023)
- [Multivariate density estimation from privatised data: universal consistency and minimax rates](https://mlsanity.com/paper/2107.12649/) (2021)
- [Private Statistical Estimation of Many Quantiles](https://mlsanity.com/paper/2302.06943/) (2023)
- [Sparsity in neural networks can improve their privacy](https://mlsanity.com/paper/2304.10553/) (2023)

## Built on by (most cited first, 2 of 2)

- [Optimal Federated Learning for Nonparametric Regression with Heterogeneous Distributed Differential Privacy Constraints](https://mlsanity.com/paper/2406.06755/) (2024)
- [Federated Nonparametric Hypothesis Testing with Differential Privacy Constraints: Optimal Rates and Adaptive Tests](https://mlsanity.com/paper/2406.06749/) (2024)

## Abstract (arXiv)

We study non-parametric density estimation for densities in Lipschitz and Sobolev spaces, and under central privacy. In particular, we investigate regimes where the privacy budget is not supposed to be constant. We consider the classical definition of central differential privacy, but also the more recent notion of central concentrated differential privacy. We recover the result of Barber and Duchi (2014) stating that histogram estimators are optimal against Lipschitz distributions for the L2 risk, and under regular differential privacy, and we extend it to other norms and notions of privacy. Then, we investigate higher degrees of smoothness, drawing two conclusions: First, and contrary to what happens with constant privacy budget (Wasserman and Zhou, 2010), there are regimes where imposing privacy degrades the regular minimax risk of estimation on Sobolev densities. Second, so-called projection estimators are near-optimal against the same classes of densities in this new setup with pure differential privacy, but contrary to the constant privacy budget case, it comes at the cost of relaxation. With zero concentrated differential privacy, there is no need for relaxation, and we prove that the estimation is optimal.