# Local differential privacy: Elbow effect in optimal density estimation and adaptation over Besov ellipsoids

arXiv: https://arxiv.org/abs/1903.01927 · HTML: https://mlsanity.com/paper/1903.01927/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Mar 2019.

## Built on by (most cited first, 13 of 13)

- [Optimal Federated Learning for Nonparametric Regression with Heterogeneous Distributed Differential Privacy Constraints](https://mlsanity.com/paper/2406.06755/) (2024)
- [Contraction of Locally Differentially Private Mechanisms](https://mlsanity.com/paper/2210.13386/) (2022)
- [Nonparametric extensions of randomized response for private confidence sets](https://mlsanity.com/paper/2202.08728/) (2022)
- [Federated Nonparametric Hypothesis Testing with Differential Privacy Constraints: Optimal Rates and Adaptive Tests](https://mlsanity.com/paper/2406.06749/) (2024)
- [Robust Estimation of Discrete Distributions under Local Differential Privacy](https://mlsanity.com/paper/2202.06825/) (2022)
- [Continual Mean Estimation Under User-Level Privacy](https://mlsanity.com/paper/2212.09980/) (2022)
- [Private Statistical Estimation of Many Quantiles](https://mlsanity.com/paper/2302.06943/) (2023)
- [Differentially Private Normalizing Flows for Density Estimation, Data Synthesis, and Variational Inference with Application to Electronic Health Records](https://mlsanity.com/paper/2302.05787/) (2023)
- [About the Cost of Central Privacy in Density Estimation](https://mlsanity.com/paper/2306.14535/) (2023)
- [Distribution-Aware Mean Estimation under User-level Local Differential Privacy](https://mlsanity.com/paper/2410.09506/) (2024)
- [Locally differentially private estimation of nonlinear functionals of discrete distributions](https://mlsanity.com/paper/2107.03940/) (2021)
- [SLDP: Semi-Local Differential Privacy for Density-Adaptive Analytics](https://mlsanity.com/paper/2602.18910/) (2026)
- [A Van Trees Lower Bound for Fully Interactive Differentially Private Federated Learning](https://mlsanity.com/paper/2605.19813/) (2026)

## Abstract (arXiv)

We address the problem of non-parametric density estimation under the additional constraint that only privatised data are allowed to be published and available for inference. For this purpose, we adopt a recent generalisation of classical minimax theory to the framework of local α-differential privacy and provide a lower bound on the rate of convergence over Besov spaces Bspq under mean integrated mathbb Lr-risk. This lower bound is deteriorated compared to the standard setup without privacy, and reveals a twofold elbow effect. In order to fulfil the privacy requirement, we suggest adding suitably scaled Laplace noise to empirical wavelet coefficients. Upper bounds within (at most) a logarithmic factor are derived under the assumption that α stays bounded as n increases: A linear but non-adaptive wavelet estimator is shown to attain the lower bound whenever p ≥ r but provides a slower rate of convergence otherwise. An adaptive non-linear wavelet estimator with appropriately chosen smoothing parameters and thresholding is shown to attain the lower bound within a logarithmic factor for all cases.