Inference Under Information Constraints III: Local Privacy Constraints

Uniformity Testing in the Shuffle Model: Simpler, Better, Faster

“…To conclude this discussion, we note that using the same "domain compression" method as [14] (as introduced in [1,3]) in a blackbox fashion, both our private-coin algorithms immediately im-ply simple public-coin robust shuffle private algorithms with sample complexity (2); actually, the slightly better…”

“…Testing uniformity of discrete distributions was first considered from the theoretical computer science viewpoint in [33], and the optimal sample complexity Θ( √ k/α 2 ) obtained in [36] -where k denotes the domain size and α the distance parameter. 1 Over the past years, several followup works refined this result, for instance to generalise it to identity testing (reference distribution other than uniform) [15,38,9,25,24,32] or to pinpoint the optimal dependence on the error probability [34,23].…”

“…The question was then revisited from the privacy perspective by [18], after which [10,12] established the tight sample complexity bound Θ( √ k/α 2 + √ k/(α √ ε) + k 1/3 /(α 4/3 ε 2/3 ) + 1/(αε)) under (central) differential privacy. In the more stringent locally private setting, the question was later raised in [37], and fully answered in a sequence of works which show that the tight sample complexity is Θ(k 3/2 /(α 2 ε 2 )) for non-interactive private-coin protocols and Θ(k/(α 2 ε 2 )) for non-interactive public-coin protocols (where the users have access to a common random seed) [2,6,1], and that allowing interactive protocols does not improve the sample complexity beyond Θ(k/(α 2 ε 2 )) [13,16,4].…”

“…This leads us to our fourth and last contribution: a locally private uniformity testing. Although optimal LDP testing algorithms are known in the high-privacy regime [2,1], for our LDP-toshuffle conversion to go through we need to start from an LDP testing algorithm with very low privacy (large ε). Building on ideas from [11], we adapt the algorithm from [2] to this low-privacy setting, obtaining as a byproduct a locally private uniformity testing algorithm with optimal sample complexity in all parameter regimes.…”

“…By definition, X := N i=1 X i = ∞ i=1 X i 1 N ≥i ; we can then compute the moment-generating function (MGF) of X as follows, using the expressions of the MGF of a Bernoulli and Poisson distributions, respectively. For t ∈ R, 1) , in which we recognize the MGF of a Poi(λp) random variable.…”