# Local Privacy, Data Processing Inequalities, and Statistical Minimax Rates

arXiv: https://arxiv.org/abs/1302.3203 · HTML: https://mlsanity.com/paper/1302.3203/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Feb 2013.

## Built on by (most cited first, 26 of 26)

- [Quantum Differential Privacy: An Information Theory Perspective](https://mlsanity.com/paper/2202.10717/) (2022)
- [When Privacy Meets Partial Information: A Refined Analysis of Differentially Private Bandits](https://mlsanity.com/paper/2209.02570/) (2022)
- [Local Differential Privacy Is Equivalent to Contraction of Eγ-Divergence](https://mlsanity.com/paper/2102.01258/) (2021)
- [Locally Differentially Private Reinforcement Learning for Linear Mixture Markov Decision Processes](https://mlsanity.com/paper/2110.10133/) (2021)
- [Quantum Local Differential Privacy and Quantum Statistical Query Model](https://mlsanity.com/paper/2203.03591/) (2022)
- [Contraction of Locally Differentially Private Mechanisms](https://mlsanity.com/paper/2210.13386/) (2022)
- [On the Statistical Complexity of Estimation and Testing under Privacy Constraints](https://mlsanity.com/paper/2210.02215/) (2022)
- [A Survey on Evaluating Quality and Trustworthiness in LLM-Generated Data](https://mlsanity.com/paper/2601.17717/) (2026)
- [Robust Estimation of Discrete Distributions under Local Differential Privacy](https://mlsanity.com/paper/2202.06825/) (2022)
- [Improved Analysis of Sparse Linear Regression in Local Differential Privacy Model](https://mlsanity.com/paper/2310.07367/) (2023)
- [Private Statistical Estimation of Many Quantiles](https://mlsanity.com/paper/2302.06943/) (2023)
- [Starlit: Privacy-Preserving Federated Learning to Enhance Financial Fraud Detection](https://mlsanity.com/paper/2401.10765/) (2024)
- [Learner-Private Convex Optimization](https://mlsanity.com/paper/2102.11976/) (2021)
- [Considerations on the Theory of Training Models with Differential Privacy](https://mlsanity.com/paper/2303.04676/) (2023)
- [About the Cost of Central Privacy in Density Estimation](https://mlsanity.com/paper/2306.14535/) (2023)
- [Enhancing Feature-Specific Data Protection via Bayesian Coordinate Differential Privacy](https://mlsanity.com/paper/2410.18404/) (2024)
- [Proactive DP: A Multple Target Optimization Framework for DP-SGD](https://mlsanity.com/paper/2102.09030/) (2021)
- [Shuffled Differentially Private Federated Learning for Time Series Data Analytics](https://mlsanity.com/paper/2307.16196/) (2023)
- [Deep Privacy Funnel Model: From a Discriminative to a Generative Approach with an Application to Face Recognition](https://mlsanity.com/paper/2404.02696/) (2024)
- [FLIPHAT: Joint Differential Privacy for High Dimensional Sparse Linear Bandits](https://mlsanity.com/paper/2405.14038/) (2024)
- [Faster Rates for Private Adversarial Bandits](https://mlsanity.com/paper/2505.21790/) (2025)
- [Minimax and Adaptive Covariance Matrix Estimation under Differential Privacy](https://mlsanity.com/paper/2603.19703/) (2026)
- [Dropout-Robust Mechanisms for Differentially Private and Fully Decentralized Mean Estimation](https://mlsanity.com/paper/2506.03746/) (2025)
- [Prediction with Expert Advice under Local Differential Privacy](https://mlsanity.com/paper/2512.06971/) (2025)
- [A Van Trees Lower Bound for Fully Interactive Differentially Private Federated Learning](https://mlsanity.com/paper/2605.19813/) (2026)
- [Unveiling the Non-Monotonic Effect of Privacy on Generalization under Byzantine Robustness](https://mlsanity.com/paper/2607.01492/) (2026)

## Abstract (arXiv)

Working under a model of privacy in which data remains private even from the statistician, we study the tradeoff between privacy guarantees and the utility of the resulting statistical estimators. We prove bounds on information-theoretic quantities, including mutual information and Kullback-Leibler divergence, that depend on the privacy guarantees. When combined with standard minimax techniques, including the Le Cam, Fano, and Assouad methods, these inequalities allow for a precise characterization of statistical rates under local privacy constraints. We provide a treatment of several canonical families of problems: mean estimation, parameter estimation in fixed-design regression, multinomial probability estimation, and nonparametric density estimation. For all of these families, we provide lower and upper bounds that match up to constant factors, and exhibit new (optimal) privacy-preserving mechanisms and computationally efficient estimators that achieve the bounds.