The Sample Complexity of Parity Learning in the Robust Shuffle Model

Abstract

Differential privacy [Dwork, McSherry, Nissim, Smith TCC 2006] is a standard of privacy in data analysis of personal information requiring that the information of any single individual should not influence the analysis’ output distribution significantly. The focus of this thesis is the shuffle model of differential privacy [Bittau, Erlingsson, Maniatis, Mironov, Raghunathan, Lie, Rudominer, Kode, Tinnés, Seefel SOSP 2017], [Cheu, Smith, Ullman, Zeber, Zhilyaev EUROCRYPT 2019]. In the shuffle model, users communicate with an analyzer via a trusted shuffler, which permutes all received messages uniformly at random before submitting them to the analyst (then the ana- lyst outputs an aggregate answer). Another model which we will discuss is the pan- privacy model [Dwork, Naor, Pitassi, Rothblum, Yekhanin ICS 2010], where an online algorithm is required to maintain differential privacy of both its internal state and its output (jointly).

We focus on the task of parity learning in the robust shuffle model and obtain the following contributions:

• We construct a reduction from a pan-private parity learner to the robust shuffle parity learner. Given an (ε, δ, 1/3)-robust shuffle private parity learner, we con- struct an (ε,δ)-pan-private parity learner. Applying recent pan-privacy lower- bounds [Cheu, Ullman 2021], we obtain a lower bound on the sample complexity Ω(2d/2) in the pan-private parity learner, which in turn implies the same lower bound in the robust shuffle model.

• We present a simple robust shuffle parity learning algorithm with sample complexity O(d2d/2). The algorithm evaluates, with differential privacy, the empirical error of all possible parity functions, and selects the one with minimal error.