Nonparametric extensions of randomized response for private confidence sets

TL;DR

This paper develops nonparametric, nonasymptotic methods for statistical inference of population means under local differential privacy, introducing private confidence intervals and sequences based on a generalized randomized response mechanism.

Contribution

It introduces a nonparametric, sequentially interactive extension of randomized response satisfying LDP, enabling private confidence intervals and sequences for means of bounded data.

Findings

Provides private analogues of Hoeffding's inequality.

Develops confidence sequences for non-stationary means.

Demonstrates private online A/B testing applications.

Abstract

This work derives methods for performing nonparametric, nonasymptotic statistical inference for population means under the constraint of local differential privacy (LDP). Given bounded observations $(X_1, \dots, X_n)$ with mean $\mu^\star$ that are privatized into $(Z_1, \dots, Z_n)$, we present confidence intervals (CI) and time-uniform confidence sequences (CS) for $\mu^\star$ when only given access to the privatized data. To achieve this, we study a nonparametric and sequentially interactive generalization of Warner's famous ``randomized response'' mechanism, satisfying LDP for arbitrary bounded random variables, and then provide CIs and CSs for their means given access to the resulting privatized observations. For example, our results yield private analogues of Hoeffding's inequality in both fixed-time and time-uniform regimes. We extend these Hoeffding-type CSs to capture time-varying (non-stationary) means, and conclude by illustrating how these methods can be used to conduct private online A/B tests.