  Differential Privacy (DP) is the current gold-standard for measuring privacy.
Estimation problems under DP constraints appearing in the literature have
largely focused on providing equal privacy to all users. We consider the
problems of empirical mean estimation for univariate data and frequency
estimation for categorical data, two pillars of data analysis in the industry,
subject to heterogeneous privacy constraints. Each user, contributing a sample
to the dataset, is allowed to have a different privacy demand. The dataset
itself is assumed to be worst-case and we study both the problems in two
different formulations -- the correlated and the uncorrelated setting. In the
former setting, the privacy demand and the user data can be arbitrarily
correlated while in the latter setting, there is no correlation between the
dataset and the privacy demand. We prove some optimality results, under both
PAC error and mean-squared error, for our proposed algorithms and demonstrate
superior performance over other baseline techniques experimentally.
