Fisher information under local differential privacy
This work addresses fundamental limits in private estimation for statisticians and data scientists, providing theoretical insights into privacy-accuracy trade-offs, but it is incremental as it builds on existing differential privacy frameworks.
The paper tackles the problem of how Fisher information scales under local differential privacy constraints, deriving bounds that show linear, quadratic, or exponential dependence on the privacy parameter ε, and applies these to achieve order-optimal lower bounds for estimation in Gaussian location and discrete distribution models.
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.