Fisher information under local differential privacy

TL;DR

This paper establishes bounds on Fisher information under local differential privacy, revealing how privacy constraints affect statistical estimation and providing optimal mechanisms for various models.

Contribution

It introduces general data processing inequalities for Fisher information under local differential privacy, applicable to diverse models and privacy levels.

Findings

Derived bounds on Fisher information scaling with privacy parameter .

Established order-optimal lower bounds for private estimation in Gaussian and discrete models.

Designed privacy mechanisms with matching squared  error in sparse Bernoulli models.

Abstract

We develop data processing inequalities that describe how Fisher information from statistical samples can scale with the privacy parameter $\varepsilon$ under local differential privacy constraints. These bounds are valid under general conditions on the distribution of the score of the statistical model, and they elucidate under which conditions the dependence on $\varepsilon$ is linear, quadratic, or exponential. We show how these inequalities imply order optimal lower bounds for private estimation for both the Gaussian location model and discrete distribution estimation for all levels of privacy $\varepsilon>0$. We further apply these inequalities to sparse Bernoulli models and demonstrate privacy mechanisms and estimators with order-matching squared $\ell^2$ error.