Testing equivalence between distributions using conditional samples

Abstract

We study a recently introduced framework [7, 8] for property testing of probability distributions, by considering distribution testing algorithms that have access to a conditional sampling oracle. This is an oracle that takes as input a subset S ⊆ [N] of the domain [N] of the unknown probability distribution D and returns a draw from the conditional probability distribution D restricted to S. This model allows considerable flexibility in the design of distribution testing algorithms; in particular, testing algorithms in this model can be adaptive.In this paper we focus on algorithms for two fundamental distribution testing problems: testing whether D = D* for an explicitly provided D*, and testing whether two unknown distributions D1 and D2 are equivalent. For both problems, the sample complexity of testing in the standard model is at least Ω(√N). For the first problem we give an algorithm in the conditional sampling model that performs only poly(1/e)-queries (for the given distance parameter e) and has no dependence on N. This improves over the poly(log N, 1/e)-query algorithm of [8]. For the second, more difficult problem, we given an algorithm whose complexity is poly(log N, 1/e). For both problems we also give efficient algorithms that work under the restriction that the algorithm perform queries only on pairs of points and provide a lower bound that is polynomial in the upper bounds.