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Abstract

We study the fundamental problems of (i) uniformity testing of a discrete distribution, and (ii) closeness testing between two discrete distributions with bounded 2-norm. These problems have been extensively studied in distribution testing and sampleoptimal estimators are known for them [17, 7, 19, 11]. In this work, we show that the original collision-based testers proposed for these problems [14, 3] are sample-optimal, up to constant factors. Previous analyses showed sample complexity upper bounds for these testers that are optimal as a function of the domain size n, but suboptimal by polynomial factors in the error parameter ε. Our main contribution is a new tight analysis establishing that these collision-based testers are information-theoretically optimal, up to constant factors, both in the dependence on n and in the dependence on ε.

Two-sample Testing of Discrete Distributions under Rare/Weak Perturbations

“…We also conduct numerical simulations to verify the theoretical analysis in finite sampling conditions. Our simulations illustrate the dominance of the minimax test over two additional tests: a test that is based on the chisquared statistic and a test that is based on the number of collisions that was popularized in works on the goodness of fit testing in the sub-linear regime [26], [12], [6], [27].…”

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“…Blais, Canonne, and Gur [26] obtain the distribution testing analogue of the communication complexity framework of [25]; and leverage it to revisit the "instance-optimal" identity testing bound of Theorem 5.8. Diakonikolas et al [46] analyze the original collision-based tester for uniformity [60], and show thatsurprisingly-it also yields optimal sample complexity (and that Poissonization, here, hurts). Finally, Jiao, Han, and Weissman [66] settle the sample complexity of tolerant testing uniformity, identity, and closeness, improving on the results of Section 5.4 with regard to the dependence on ε 2 − ε 1 .…”

“…Furthermore, for distributions on Ω = [n] any EVAL D query can be simulated by (at most) two queries to a CEVAL D oracle, that is, the cumulative dual model is at least as powerful as the dual one. 46 Remark 12.4 (On the relation to p -testing for functions on the line). We note that testing distributions with EVAL D access is strongly reminiscent of the recent results of Berman et al [21] on testing functions f : [n] → [0, 1] with relation to p distances.…”

“…First, the distance they consider is normalized (by a factor n in the case of 1 distance), so that a straightforward translation between the two settings would imply 45 To the best of our knowledge, such cumulative evaluation oracle CEVAL appears for the first time in [20,Section 8]. 46 We mention that Canonne and Rubinfeld discuss in [35] a relaxation of these two models, where the queries to the corresponding evaluation oracles are only answered within a multiplicative (1 ± γ) factor. They observe that many of their algorithms can be made robust against such multiplicative noise, while maintaining their query complexity (see, e. g., Table 1 of [35]).…”