Conference Agenda

In this paper, we study the problem of estimating the unknown mean $\theta$ of a unit variance Gaussian distribution in a locally differentially private (LDP) way. In the high-privacy regime ($\epsilon\le 1$), we identify an optimal privacy mechanism that minimizes the variance of the estimator asymptotically. Our main technical contribution is the maximization of the Fisher-Information of the sanitized data with respect to the local privacy mechanism $Q$. We find that the exact solution $Q_{\theta,\epsilon}$ of this maximization is the sign mechanism that applies randomized response to the sign of $X_i-\theta$, where $X_1,\dots, X_n$ are the confidential iid original samples. However, since this optimal local mechanism depends on the unknown mean $\theta$, we employ a two-stage LDP parameter estimation procedure which requires splitting agents into two groups. The first $n_1$ observations are used to consistently but not necessarily efficiently estimate the parameter $\theta$ by $\tilde{\theta}_{n_1}$. Then this estimate is updated by applying the sign mechanism with $\tilde{\theta}_{n_1}$ instead of $\theta$ to the remaining $n-n_1$ observations, to obtain an LDP and efficient estimator of the unknown mean.

We study the task of state estimation and quantum hypothesis testing under the constraint of gentle measurements acting independently on $n$ independent identical states. Gentle measurements are quantum instruments for which the post-measurement state $\rho(\theta)_{M \to y}$ of a state $\rho(\theta)$ differs only by a small amount $\alpha$ from the pre-measurement state, i.e. $|| \rho(\theta)_{M_\to y} - \rho(\theta)||_1 \leq \alpha$. We show that both the testing and estimation errors scale with a factor of $\frac{1}{\alpha \sqrt{n}}$ which is consistent with the relation established between gentleness and local differential privacy.

This work is concerned with the problem of estimating the $m$-fold convolution of the densities of $m$ independent variables in the first step and vectors in the second step. A nonparametric estimator is proposed and the point-wise and integrated quadratic risk is studied. We use Fourier analysis to bound the variance and kernel methods allow us to consider Nikolski and Hölder classes in addition to the standard Sobolev classes for deconvolution estimators. For this, smoothness properties of $m$-fold convoluted densities are studied. We discuss rates of convergence. In addition, we look at bandwidth selection methods.

We consider the nonparametric estimation of a function of interest $\theta$ based on empirical versions of both an observable signal $g=s\,\theta$ and a multiplicative function $s$. The general framework covers, for example, circular convolution, additive convolution on the real line, and multiplicative convolution on the positive real line. Typical questions in this context are the nonparametric estimation of the function $\theta$ as a whole or the value of a linear functional evaluated at $\theta$, referred to as global or local estimation, respectively. The proposed estimation procedures necessitates the choice of a tuning parameter, which in turn, crucially influences the attainable accuracy of the constructed estimator. Its optimal choice, however, follows from a classical squared-bias-variance trade-off and relies on an a-priori knowledge about $\theta$ and $s$, which is usually inaccessible in practice. We propose a fully data-driven choice of the tuning parameter by model selection and Goldenshluger-Lepski method for global and local estimation, respectively. We derive global and local oracle inequalities and discuss attainable minimax rates of convergence considering usual behaviours for $\theta$ and $s$.