Distributed Uniformity Testing

Fischer, Orr; Meir, Uri; Oshman, Rotem

doi:10.1145/3212734.3212772

Cited by 10 publications

(24 citation statements)

References 10 publications

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“…However, in that paper the authors consider a blackboard model of communication and strive to minimize the total number of bits communicated, without placing any restriction on the number of bits per sample. A variant of the distribution testing problem is considered in [24] where players observe multiple samples and communicate their local test results to the central referee who is required to use simple aggregation rules such as AND. Interestingly, such setups have received a lot of attention in the sensor network literature where a fusion center combines local decisions using simple rules such as majority; see [49] for an early review.…”

Section: Prior Workmentioning

confidence: 99%

Inference under Information Constraints I: Lower Bounds from Chi-Square Contraction

Acharya¹,

Canonne²,

Tyagi³

2018

Preprint

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Multiple players are each given one independent sample, about which they can only provide limited information to a central referee. Each player is allowed to describe its observed sample to the referee using a channel from a family of channels W, which can be instantiated to capture both the communicationand privacy-constrained settings and beyond. The referee uses the messages from players to solve an inference problem for the unknown distribution that generated the samples. We derive lower bounds for sample complexity of learning and testing discrete distributions in this information-constrained setting.Underlying our bounds is a characterization of the contraction in chi-square distances between the observed distributions of the samples when information constraints are placed. This contraction is captured in a local neighborhood in terms of chi-square and decoupled chi-square fluctuations of a given channel, two quantities we introduce. The former captures the average distance between distributions of channel output for two product distributions on the input, and the latter for a product distribution and a mixture of product distribution on the input. Our bounds are tight for both public-and privatecoin protocols. Interestingly, the sample complexity of testing is order-wise higher when restricted to private-coin protocols.

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Section: Prior Workmentioning

confidence: 99%

Inference under Information Constraints I: Lower Bounds from Chi-Square Contraction

Acharya¹,

Canonne²,

Tyagi³

2018

Preprint

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“…Lastly, in Section 3.3, we use the structural theorem to show how on certain graphs one can solve uniformity in the CONGEST model faster than what was previously known to be possible. Specifically, if the communication network has k players and diameter D, and each players start with one sample, the best known algorithm runs in O(D + n/(k 4 )) rounds [17]. The improvement we suggest is an algorithm that takes O(D) rounds, and works in specific networks that have a certain topological characteristics.…”

Section: Models and Resultsmentioning

confidence: 99%

“…Another interesting graph (used in some sense in [17]) is actually a union of disjoint cliques. This graph turns out to be quite useful.…”

Section: Disjoint Cliquesmentioning

confidence: 99%

“…In the past few years there has been a growing interest in distribution testing under various computational models, including collaborative testing in multiparty model, the streaming model, privacy aspects of testing, and others ( [6,7,17,9,2,23,4,5,1,24,18,8] and more). We mention in more details the works concerning uniformity testers in models we pursue.…”

Section: Related Workmentioning

confidence: 99%

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Comparison Graphs: a Unified Method for Uniformity Testing

Meir

2020

Preprint

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Distribution testing can be described as follows: q samples are being drawn from some unknown distribution P over a known domain [n]. After the sampling process, a decision must be made about whether P holds some property, or is far from it. The most studied problem in the field is arguably uniformity testing, where one needs to distinguish the case that P is uniform over [n] from the case that P is -far from being uniform (in 1). The sample complexity of a property is the amount of necessary and sufficient samples to test for this property, and it is known to be Θ √ n/ 2 when testing for uniformity. This problem was recently considered in various restricted models that pose, for example, communication or memory constraints. In more than one occasion, the known optimal solution boils down to counting collisions among the drawn samples (each two samples that have the same value add one to the count). This idea dates back to the first uniformity tester, and was coined the name "collision-based tester".In this paper, we introduce the notion of comparison graphs and use it to formally define a generalized collision-based tester. Roughly speaking, the edges of the graph indicate the tester which pairs of samples should be compared (that is, the original tester is induced by a clique, where all pairs are being compared). We prove a structural theorem that gives a sufficient condition for a comparison graph to induce a good uniformity tester. As an application, we develop a generic method to test uniformity, and devise nearly-optimal uniformity testers under various computational constraints. We improve and simplify a few known results, and introduce a new model in which the method also produces an efficient tester.The idea behind our method is to translate the computational constraints of a certain model to ones on the comparison graph, which paves the way to finding a good graph: a set of pairs that can be compared in this model, and induces a uniformity tester. We believe that in future consideration of uniformity testing in new models, our method can be used to obtain efficient testers with minimal effort.

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“…Recently, there has been a surge of interest in distributed statistical inference, focusing on density or parameter estimation under communication constraints [HMÖW18b,HÖW18,HMÖW18a,BHÖ19] or local privacy [DJW17, EPK14, YB18, KBR16, ASZ19, AS19]. The testing counterpart, specifically identity testing, was studied in the locally differentially private (LDP) setting by Gaboardi and Rogers [GR18] and Sheffet [She18], followed by [ACFT19]; and in the communication-constrained setting in [ACT19c,ACT19b], as well as by (with a slightly different focus) [FMO18]. The role of public randomness in distributed testing was explicitly studied in [ACT19c,ACT19b], which showed a quantitative gap between the sample complexities of public-and private-coin protocols; those works, however, left open the fine-grained question of limited public randomness we study here.…”

Section: Prior and Related Workmentioning

confidence: 99%

Domain Compression and its Application to Randomness-Optimal Distributed Goodness-of-Fit

Acharya¹,

Canonne²,

Sun³

et al. 2019

Preprint

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We study goodness-of-fit of discrete distributions in the distributed setting, where samples are divided between multiple users who can only release a limited amount of information about their samples due to various information constraints. Recently, a subset of the authors showed that having access to a common random seed (i.e., shared randomness) leads to a significant reduction in the sample complexity of this problem. In this work, we provide a complete understanding of the interplay between the amount of shared randomness available, the stringency of information constraints, and the sample complexity of the testing problem by characterizing a tight trade-off between these three parameters. We provide a general distributed goodness-of-fit protocol that as a function of the amount of shared randomness interpolates smoothly between the private-and public-coin sample complexities. We complement our upper bound with a general framework to prove lower bounds on the sample complexity of this testing problems under limited shared randomness. Finally, we instantiate our bounds for the two archetypal information constraints of communication and local privacy, and show that our sample complexity bounds are optimal as a function of all the parameters of the problem, including the amount of shared randomness.A key component of our upper bounds is a new primitive of domain compression, a tool that allows us to map distributions to a much smaller domain size while preserving their pairwise distances, using a limited amount of randomness.

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Distributed Uniformity Testing

Cited by 10 publications

References 10 publications

Inference under Information Constraints I: Lower Bounds from Chi-Square Contraction

Inference under Information Constraints I: Lower Bounds from Chi-Square Contraction

Comparison Graphs: a Unified Method for Uniformity Testing

Domain Compression and its Application to Randomness-Optimal Distributed Goodness-of-Fit

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