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

## TL;DR

This paper investigates the limits of non-parametric density estimation under local differential privacy constraints, revealing an elbow effect in convergence rates and proposing wavelet-based estimators that achieve near-optimal performance.

## Contribution

It introduces a lower bound on estimation rates under local differential privacy and develops wavelet estimators that adaptively attain these bounds across Besov spaces.

## Key findings

- Lower bounds show deterioration of convergence rates due to privacy constraints.
- A wavelet estimator attains the lower bound when p ≥ r.
- An adaptive wavelet estimator achieves near-optimal rates in all cases.

## Abstract

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 $\alpha$-differential privacy and provide a lower bound on the rate of convergence over Besov spaces $B^s_{pq}$ under mean integrated $\mathbb L^r$-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 $\alpha$ stays bounded as $n$ increases: A linear but non-adaptive wavelet estimator is shown to attain the lower bound whenever $p \geq r$ 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.