Minimax Rates of Estimating Approximate Differential Privacy

TL;DR

This paper develops a data-driven method to accurately estimate approximate differential privacy guarantees, reducing errors in privacy mechanism verification through a novel polynomial approximation estimator with optimal sample complexity.

Contribution

It introduces a minimax optimal estimator for privacy guarantees using polynomial approximation, achieving effective sample size amplification and providing fundamental lower bounds.

Findings

Estimator achieves near-optimal performance with $n \,\ln\, n$ samples

Proves minimax optimality via matching lower bounds

Demonstrates effective sample size amplification phenomenon

Abstract

Differential privacy has become a widely accepted notion of privacy, leading to the introduction and deployment of numerous privatization mechanisms. However, ensuring the privacy guarantee is an error-prone process, both in designing mechanisms and in implementing those mechanisms. Both types of errors will be greatly reduced, if we have a data-driven approach to verify privacy guarantees, from a black-box access to a mechanism. We pose it as a property estimation problem, and study the fundamental trade-offs involved in the accuracy in estimated privacy guarantees and the number of samples required. We introduce a novel estimator that uses polynomial approximation of a carefully chosen degree to optimally trade-off bias and variance. With $n$ samples, we show that this estimator achieves performance of a straightforward plug-in estimator with $n \ln n$ samples, a phenomenon referred to as effective sample size amplification. The minimax optimality of the proposed estimator is proved by comparing it to a matching fundamental lower bound.