A Chasm Between Identity and Equivalence Testing with Conditional Queries

Abstract

A recent model for property testing of probability distributions [CFGM13, CRS15] 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 [CRS15] 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 √ 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 [CRS15] established a polylog(n)-query upper bound for equivalence testing, very recently brought down to O(log logn) by Falahatgar et al. [FJO+15], 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 [Sublinear.info, Problem 66]. In this work, we answer the question in the positive, showing that any testing algorithm for equivalence must make Ω (√ log logn ) 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 nΘ(1)). Turning to another question, we investigate the complexity of support size estimation. We provide a doubly-logarithmic upper bound for the adaptive version of this problem, generalizing work of Ron and Tsur [RT14] to our weaker model. We also establish a logarithmic lower bound for the non-adaptive version of this problem. This latter result carries on to the related problem of non-adaptive uniformity testing, an exponential improvement over previous results that resolves an open question of Chakraborty, Fischer, Goldhirsh, and Matsliah [CFGM13]. ∗EECS, MIT. Email: jayadev@csail.mit.edu. Research supported by grant from MITEI-Shell program. †Columbia University. Email: ccanonne@cs.columbia.edu. Research supported by NSF CCF-1115703 and NSF CCF-1319788. ‡EECS, MIT. Email: g@csail.mit.edu.