Differential Privacy for Sequential Algorithms

Abstract

We study the differential privacy of sequential statistical inference and learning algorithms that are characterized by random termination time. Using the two examples: sequential probability ratio test and sequential empirical risk minimization, we show that the number of steps such algorithms execute before termination can jeopardize the differential privacy of the input data in a similar fashion as their outputs, and it is impossible to use the usual Laplace mechanism to achieve standard differentially private in these examples. To remedy this, we propose a notion of weak differential privacy and demonstrate its equivalence to the standard case for large i.i.d. samples. We show that using the Laplace mechanism, weak differential privacy can be achieved for both the sequential probability ratio test and the sequential empirical risk minimization with proper performance guarantees. Finally, we provide preliminary experimental results on the Breast Cancer Wisconsin (Diagnostic) and Landsat Satellite Data Sets from the UCI repository. 1

A nonparametric extension of randomized response for private confidence sets

“…The literature on differentially private statistical inference is rich, including nonparametric estimation rates, [14,5,15,6,16,17], parametric hypothesis testing and confidence intervals, [18,19,20,21,22,23,24,25,26] median estimation, [27], independence testing [28], online convex optimization [29], and parametric sequential hypothesis testing [30]. A more detailed summary of these prior works can be found in Section 6…”

“…Figure 3: Widths of private 90%-CIs for the mean of a Beta (10,30) distribution satisfying ε-LDP with ε " 2. This is a low-variance distribution, so Hoeffding-type methods (NPRR-H-CI, Lap-H-CI) do slightly worse than the variance-adaptive NPRR-pmKelly-CI for large n. Theorem 3.2 (NPRR-H-CI).…”

“…Moreover, like most of the statistics literature, the prior work on private statistical inference is non-sequential, with the exception of Wang et al [30] who study private analogues of Wald's sequential probability ratio test [48] for simple hypotheses. Another interesting paper is that of Jun and Orabona [29,Sections 7.1 & 7.2].…”

“…[44, 45], combined with Proposition 3.7. Similarly, a two-sided CS can be obtained by replacing p ϕ t ´r B t in(30) with p ϕ t ˘r B t , where r B t is given in(18).…”

Private Sequential Hypothesis Testing for Statisticians: Privacy, Error Rates, and Sample Size

“…Our work deals with the sequential hypothesis testing problem which is essentially a classification problem, and our aim is to provide a unifying approach by showing that a generalization of this technique can be applied to solve general private sequential hypothesis testing problems for a more general class of accuracy objectives. [WSMD20] considers privatization of SPRT. Their algorithm is to add Laplace noise to the thresholds to generate a noisy stopping time, and then use exponential mechanism to output the binary decision.…”