We provide improved differentially private algorithms for identity testing of high-dimensional distributions. Specifically, for d-dimensional Gaussian distributions with known covariance Σ, we can test whether the distribution comes from N(μ∗,Σ) for some fixed μ∗ or from some N(μ,Σ) with total variation distance at least α from N(μ∗,Σ) with (ε,0)-differential privacy, using only
O~(d1/2α2+d1/3α4/3⋅ε2/3+1α⋅ε)
samples if the algorithm is allowed to be computationally inefficient, and only
O~(d1/2α2+d1/4α⋅ε)
samples for a computationally efficient algorithm. We also provide a matching lower bound showing that our computationally inefficient algorithm has optimal sample complexity. We also extend our algorithms to various related problems, including mean testing of Gaussians with bounded but unknown covariance, uniformity testing of product distributions over {−1,1}d, and tolerant testing.
Our results improve over the previous best work of Canonne et al.~\cite{CanonneKMUZ20} for both computationally efficient and inefficient algorithms, and even our computationally efficient algorithm matches the optimal \emph{non-private} sample complexity of O(d√α2) in many standard parameter settings.
In addition, our results show that, surprisingly, private identity testing of d-dimensional Gaussians can be done with fewer samples than private identity testing of discrete distributions over a domain of size d \cite{AcharyaSZ18}, which refutes a conjectured lower bound of~\cite{CanonneKMUZ20}.
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Differentially Private Testing of Identity and Closeness of Discrete Distributions
We study the fundamental problems of identity testing (goodness of fit), and closeness testing (two sample test) of distributions over k elements, under differential privacy. While the problems have a long history in statistics, finite sample bounds for these problems have only been established recently.
In this work, we derive upper and lower bounds on the sample complexity of both the problems under (epsilon, delta)-differential privacy. We provide sample optimal algorithms for identity testing problem for all parameter ranges, and the first results for closeness testing. Our closeness testing bounds are optimal in the sparse regime where the number of samples is at most k.
Our upper bounds are obtained by privatizing non-private estimators for these problems. The non-private estimators are chosen to have small sensitivity. We propose a general framework to establish lower bounds on the sample complexity of statistical tasks under differential privacy. We show a bound on di erentially private algorithms in terms of a coupling between the two hypothesis classes we aim to test. By carefully constructing chosen priors over the hypothesis classes, and using Le Cam’s two point theorem we provide a general mechanism for proving lower bounds. We believe that the framework can be used to obtain strong lower bounds for other statistical tasks under privacy.
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- Award ID(s):
- 1657471
- NSF-PAR ID:
- 10101015
- Date Published:
- Journal Name:
- Advances in Neural Information Processing Systems 31 (NIPS 2018)
- Volume:
- 31
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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