A Chasm Between Identity and Equivalence Testing with Conditional Queries

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 Omega(root 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(log n)-query upper bound for equivalence testing, very recently brought down to (O) over tilde (log log n) 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 Omega(root log log n) 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(Theta(1))). We also obtain results on the query complexity of uniformity testing and support-size estimation with conditional samples. In particular, we answer a question of Chakraborty et al. (ITCS 2013) showing that non-adaptive uniformity testing indeed requires Omega(log n) queries in the conditional model. This is an exponential improvement on their previous lower bound of Omega(log log n), and matches up to polynomial factors their poly(log n) upper bound. For the problem of support-size estimation, we provide both adaptive and non -adaptive algorithms, with query complexities poly(log log n) and poly(log n), respectively, and complement them with a lower bound of Omega(log n) conditional queries for non-adaptive algorithms.