文章基本信息

标题：A Chasm Between Identity and Equivalence Testing with Conditional Queries
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作者：Jayadev Acharya ; Clément L. Canonne ; Gautam Kamath 等
期刊名称：Theory of Computing
印刷版ISSN：1557-2862
电子版ISSN：1557-2862
出版年度：2018
卷号：14
页码：1-46
DOI：10.4086/toc.2018.v014a019
出版社：University of Chicago
摘要：A recent model for property testing of probability distributions (Chakraborty et al., ITCS 2013, Canonne et al., SICOMP 2015) enables tremendous savings in the sample complexity of testing algorithms, by allowing them to condition the sampling on subsets of the domain. In particular, Canonne, Ron, and Servedio (SICOMP 2015) showed that, in this setting, testing identity of an unknown distribution D (i. e., whether D = D ∗ for an explicitly known D ∗ ) can be done with a constant number of queries (i. e., samples), independent of the support size n—in contrast to the required Ω( √ n) in the standard sampling model. However, it was unclear whether the same stark contrast exists for the case of testing equivalence, where both distributions are unknown. Indeed, while Canonne et al. established a poly(logn)- query upper bound for equivalence testing, very recently brought down to O˜(loglogn) by Falahatgar et al. (COLT 2015), whether a dependence on the domain size n is necessary was still open, and explicitly posed by Fischer at the Bertinoro Workshop on Sublinear Algorithms (2014). In this article, we answer the question in the affirmative, showing that any testing algorithm for equivalence must make Ω √ loglogn queries in the conditional sampling model. Interestingly, this demonstrates a gap between identity and equivalence testing, absent in the standard sampling model (where both problems have sampling complexity n Θ(1) ).
关键词：property testing; distribution testing; conditional sampling; lower bounds;; equivalence testing; uniformity testing; support size estimation