The Uniform Distribution Is Complete with Respect to Testing Identity to a Fixed Distribution

Goldreich, Oded

doi:10.1007/978-3-030-43662-9_10

Cited by 32 publications

(46 citation statements)

References 9 publications

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“…Remark. Uniformity testing has been a useful algorithmic primitive for several other distribution testing problems as well [2,8,11,10,6,12]. Notably, Goldreich [12] recently showed that the more general problem of testing the identity of any explicitly given distribution can be reduced to uniformity testing with only a constant factor loss in sample complexity.…”

Section: Background and Our Resultsmentioning

confidence: 99%

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Diakonikolas¹,

Gouleakis²,

Peebles³

et al. 2019

Chicago J. of Theoretical Comp. Sci.

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We study the fundamental problems of (i) uniformity testing of a discrete distribution, and (ii) closeness testing between two discrete distributions with bounded 2-norm. These problems have been extensively studied in distribution testing and sampleoptimal estimators are known for them [17, 7, 19, 11]. In this work, we show that the original collision-based testers proposed for these problems [14, 3] are sample-optimal, up to constant factors. Previous analyses showed sample complexity upper bounds for these testers that are optimal as a function of the domain size n, but suboptimal by polynomial factors in the error parameter ε. Our main contribution is a new tight analysis establishing that these collision-based testers are information-theoretically optimal, up to constant factors, both in the dependence on n and in the dependence on ε.

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Section: Background and Our Resultsmentioning

confidence: 99%

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Diakonikolas¹,

Gouleakis²,

Peebles³

et al. 2019

Chicago J. of Theoretical Comp. Sci.

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“…Following the first version of this survey, several works have been published which settle or address some of the problems covered in this chapter; we hereafter mention a few of them. Diakonikolas and Kane [47] provide a new framework to prove upper bounds for a variety of distribution testing problems, essentially by an elegant reduction from 1 to 2 testing (see also [58] for an exposition), as well as an information-theoretic framework for establishing lower bounds. Canonne [29] proves near-tight upper and lower bounds for the problem of testing the class of k-histograms discussed in Section 6.2.…”

Section: Subsequent Workmentioning

confidence: 99%

“…The Poisson distribution has many key properties: amongst others, the sum of finitely many Poisson random variables is itself Poisson; a Poisson random variable is tightly concentrated around its expectation; and the Poisson distribution can be viewed as the limit of a Binomial distribution Bin(n, p) when n goes to infinity while keeping the product λ = np constant. 58 Fact D.10. Let Ω be a discrete domain, and D ∈ ∆(Ω).…”

Section: D3 Poissonizationmentioning

confidence: 99%

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Canonne

2020

Theory of Comput.

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The field of property testing originated in work on program checking, and has evolved into an established and very active research area. In this work, we survey the developments of one of its most recent and prolific offsprings, distribution testing. This subfield, at the junction of property testing and Statistics, is concerned with studying properties of probability distributions.We cover the current status of distribution testing in several settings, starting with the traditional sampling model where the algorithm obtains independent samples from the distribution. We then discuss different recent models, which either grant the testing algorithms more powerful types of queries, or evaluate their performance against that of an information-theoretically optimal "adversary" (for a given number of samples). In each setting, we describe the state of the art for a variety of testing problems.We hope this survey will serve as a self-contained introduction for those considering research in this field.

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“…A first question is to sharply characterize the complexity of general identity testing. While in the SAMP model, uniformity testing is known to be complete for identity testing [Gol16], a moment's thought indicates that the same reduction does not immediately hold for either the COND or NACOND model. This is (roughly) because Goldreich's reduction involves mapping the problem onto a larger domain, which would require more "granular" conditional samples than afforded by standard conditional sampling models in order for the reduction to go through.…”

Section: Open Problemsmentioning

confidence: 99%

“…We note that the SAMP-model reduction of[Gol16], from identity testing to uniformity testing, is not known to apply in either the NACOND or COND models.…”

mentioning

confidence: 99%

Anaconda: A Non-Adaptive Conditional Sampling Algorithm for Distribution Testing

Kamath¹,

Tzamos²

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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We investigate distribution testing with access to non-adaptive conditional samples. In the conditional sampling model, the algorithm is given the following access to a distribution: it submits a query set S to an oracle, which returns a sample from the distribution conditioned on being from S. In the non-adaptive setting, all query sets must be specified in advance of viewing the outcomes.Our main result is the first polylogarithmic-query algorithm for equivalence testing, deciding whether two unknown distributions are equal to or far from each other. This is an exponential improvement over the previous best upper bound, and demonstrates that the complexity of the problem in this model is intermediate to the the complexity of the problem in the standard sampling model and the adaptive conditional sampling model. We also significantly improve the sample complexity for the easier problems of uniformity and identity testing. For the former, our algorithm requires onlyÕ(log n) queries, matching the information-theoretic lower bound up to a O(log log n)-factor.Our algorithm works by reducing the problem from ℓ1-testing to ℓ∞-testing, which enjoys a much cheaper sample complexity. Necessitated by the limited power of the non-adaptive model, our algorithm is very simple to state. However, there are significant challenges in the analysis, due to the complex structure of how two arbitrary distributions may differ.

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The Uniform Distribution Is Complete with Respect to Testing Identity to a Fixed Distribution

Cited by 32 publications

References 9 publications

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Anaconda: A Non-Adaptive Conditional Sampling Algorithm for Distribution Testing

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