# Local Differential Privacy Is Equivalent to Contraction of Eγ-Divergence

arXiv: https://arxiv.org/abs/2102.01258 · HTML: https://mlsanity.com/paper/2102.01258/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 45 references that are in the map.
- First posted on arXiv 2 Feb 2021.

## Builds on (most cited first, 13 of 13)

- [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)
- [Privacy Amplification by Subsampling: Tight Analyses via Couplings and Divergences](https://mlsanity.com/paper/1807.01647/) (2018)
- [Concentrated Differential Privacy](https://mlsanity.com/paper/1603.01887/) (2016)
- [Protection Against Reconstruction and Its Applications in Private Federated Learning](https://mlsanity.com/paper/1812.00984/) (2018)
- [Discrete Distribution Estimation under Local Privacy](https://mlsanity.com/paper/1602.07387/) (2016)
- [Inference under Information Constraints I: Lower Bounds from Chi-Square Contraction](https://mlsanity.com/paper/1812.11476/) (2018)
- [Optimal Schemes for Discrete Distribution Estimation under Locally Differential Privacy](https://mlsanity.com/paper/1702.00610/) (2017)
- [Local Privacy, Data Processing Inequalities, and Statistical Minimax Rates](https://mlsanity.com/paper/1302.3203/) (2013)
- [Three Variants of Differential Privacy: Lossless Conversion and Applications](https://mlsanity.com/paper/2008.06529/) (2020)
- [Fisher information under local differential privacy](https://mlsanity.com/paper/2005.10783/) (2020)
- [Privacy Amplification by Mixing and Diffusion Mechanisms](https://mlsanity.com/paper/1905.12264/) (2019)
- [Locally Private Mean Estimation: Z-test and Tight Confidence Intervals](https://mlsanity.com/paper/1810.08054/) (2018)

## Built on by (most cited first, 17 of 17)

- [Quantum Differential Privacy: An Information Theory Perspective](https://mlsanity.com/paper/2202.10717/) (2022)
- [Optimal and Differentially Private Data Acquisition: Central and Local Mechanisms](https://mlsanity.com/paper/2201.03968/) (2022)
- [The Sample Complexity of Approximate Rejection Sampling with Applications to Smoothed Online Learning](https://mlsanity.com/paper/2302.04658/) (2023)
- [Quantum Local Differential Privacy and Quantum Statistical Query Model](https://mlsanity.com/paper/2203.03591/) (2022)
- [Federated Linear Contextual Bandits with User-level Differential Privacy](https://mlsanity.com/paper/2306.05275/) (2023)
- [Contraction of Locally Differentially Private Mechanisms](https://mlsanity.com/paper/2210.13386/) (2022)
- [Contraction of Private Quantum Channels and Private Quantum Hypothesis Testing](https://mlsanity.com/paper/2406.18651/) (2024)
- [Auditing Differential Privacy Guarantees Using Density Estimation](https://mlsanity.com/paper/2406.04827/) (2024)
- [Characterizing Dependence of Samples along the Langevin Dynamics and Algorithms via Contraction of Phi-Mutual Information](https://mlsanity.com/paper/2402.17067/) (2024)
- [Lower Bounds on the Bayesian Risk via Information Measures](https://mlsanity.com/paper/2303.12497/) (2023)
- [Measured Hockey-Stick Divergence and its Applications to Quantum Pufferfish Privacy](https://mlsanity.com/paper/2501.12359/) (2025)
- [Deep Privacy Funnel Model: From a Discriminative to a Generative Approach with an Application to Face Recognition](https://mlsanity.com/paper/2404.02696/) (2024)
- [High-Probability Bounds For Heterogeneous Local Differential Privacy](https://mlsanity.com/paper/2510.11895/) (2025)
- [Differentially Private Secure Multiplication: Hiding Information in the Rubble of Noise](https://mlsanity.com/paper/2309.16105/) (2023)
- [Signal Recovery from Random Dot-Product Graphs Under Local Differential Privacy](https://mlsanity.com/paper/2504.17274/) (2025)
- [Tighter Information-Theoretic Generalization Bounds via a Novel Class of Change of Measure Inequalities](https://mlsanity.com/paper/2602.07999/) (2026)
- [Privacy-Utility Tradeoffs in Quantum Information Processing](https://mlsanity.com/paper/2602.10510/) (2026)

## Abstract (arXiv)

We investigate the local differential privacy (LDP) guarantees of a randomized privacy mechanism via its contraction properties. We first show that LDP constraints can be equivalently cast in terms of the contraction coefficient of the Eγ-divergence. We then use this equivalent formula to express LDP guarantees of privacy mechanisms in terms of contraction coefficients of arbitrary f-divergences. When combined with standard estimation-theoretic tools (such as Le Cam's and Fano's converse methods), this result allows us to study the trade-off between privacy and utility in several testing and minimax and Bayesian estimation problems.