Local Privacy and Minimax Bounds: Sharp Rates for Probability Estimation

TL;DR

This paper establishes sharp minimax rates for probability distribution estimation under local privacy constraints, revealing fundamental tradeoffs and demonstrating the optimality of Warner's randomized response method.

Contribution

It provides the first precise minimax bounds for private probability estimation and introduces tools to balance privacy and statistical efficiency.

Findings

Sharp minimax convergence rates under local privacy

Tradeoffs between privacy level and estimation accuracy

Warner's randomized response is optimal for survey privacy

Abstract

We provide a detailed study of the estimation of probability distributions---discrete and continuous---in a stringent setting in which data is kept private even from the statistician. We give sharp minimax rates of convergence for estimation in these locally private settings, exhibiting fundamental tradeoffs between privacy and convergence rate, as well as providing tools to allow movement along the privacy-statistical efficiency continuum. One of the consequences of our results is that Warner's classical work on randomized response is an optimal way to perform survey sampling while maintaining privacy of the respondents.