Abstract
A recent model for property testing of probability distributions 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 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 samples, independent of the support size n  in contrast to the required sqrt(n) in the standard sampling model. However, it was unclear whether the same held for the case of testing equivalence, where both distributions are unknown. Indeed, while Canonne, Ron, and Servedio established a polylog(n)query upper bound for equivalence testing, very recently brought down to ~O(log(log(n))) by Falahatgar et al., 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. In this work, we answer the question in the positive, showing that any testing algorithm for equivalence must make Omega(sqrt(log(log(n)))) queries in the conditional sampling model. Interestingly, this demonstrates an intrinsic qualitative gap between identity and equivalence testing, absent in the standard sampling model (where both problems have sampling complexity n^(Theta(1))).
Turning to another question, we investigate the complexity of support size estimation. We provide a doublylogarithmic upper bound for the adaptive version of this problem, generalizing work of Ron and Tsur to our weaker model. We also establish a logarithmic lower bound for the nonadaptive version of this problem. This latter result carries on to the related problem of nonadaptive uniformity testing, an exponential improvement over previous results that resolves an open question of Chakraborty, Fischer, Goldhirsh, and Matsliah.
BibTeX  Entry
@InProceedings{acharya_et_al:LIPIcs:2015:5317,
author = {Jayadev Acharya and Cl{\'e}ment L. Canonne and Gautam Kamath},
title = {{A Chasm Between Identity and Equivalence Testing with Conditional Queries}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
pages = {449466},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897897},
ISSN = {18688969},
year = {2015},
volume = {40},
editor = {Naveen Garg and Klaus Jansen and Anup Rao and Jos{\'e} D. P. Rolim},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2015/5317},
URN = {urn:nbn:de:0030drops53178},
doi = {10.4230/LIPIcs.APPROXRANDOM.2015.449},
annote = {Keywords: property testing, probability distributions, conditional samples}
}
Keywords: 

property testing, probability distributions, conditional samples 
Collection: 

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015) 
Issue Date: 

2015 
Date of publication: 

13.08.2015 