We study the problem of testing identity against a given distribution with a focus on the high confidence regime. More precisely, given samples from an unknown distribution p over n elements, an explicitly given distribution q, and parameters 0< epsilon, delta < 1, we wish to distinguish, with probability at least 1delta, whether the distributions are identical versus epsilonfar in total variation distance. Most prior work focused on the case that delta = Omega(1), for which the sample complexity of identity testing is known to be Theta(sqrt{n}/epsilon^2). Given such an algorithm, one can achieve arbitrarily small values of delta via blackbox amplification, which multiplies the required number of samples by Theta(log(1/delta)). We show that blackbox amplification is suboptimal for any delta = o(1), and give a new identity tester that achieves the optimal sample complexity. Our new upper and lower bounds show that the optimal sample complexity of identity testing is Theta((1/epsilon^2) (sqrt{n log(1/delta)} + log(1/delta))) for any n, epsilon, and delta. For the special case of uniformity testing, where the given distribution is the uniform distribution U_n over the domain, our new tester is surprisingly simple: to test whether p = U_n versus d_{TV} (p, U_n) >= epsilon, we simply threshold d_{TV}({p^}, U_n), where {p^} is the empirical probability distribution. The fact that this simple "plugin" estimator is sampleoptimal is surprising, even in the constant delta case. Indeed, it was believed that such a tester would not attain sublinear sample complexity even for constant values of epsilon and delta. An important contribution of this work lies in the analysis techniques that we introduce in this context. First, we exploit an underlying strong convexity property to bound from below the expectation gap in the completeness and soundness cases. Second, we give a new, fast method for obtaining provably correct empirical estimates of the true worstcase failure probability for a broad class of uniformity testing statistics over all possible input distributions  including all previously studied statistics for this problem. We believe that our novel analysis techniques will be useful for other distribution testing problems as well.
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Approximating the Distance to Monotonicity of Boolean Functions
We design a nonadaptive algorithm that, given a Boolean function f: {0, 1}^n → {0, 1} which is αfar from monotone, makes poly(n, 1/α) queries and returns an estimate that, with high probability, is an Otilde(\sqrt{n})approximation to the distance of f to monotonicity. Furthermore, we show that for any constant k > 0, approximating the distance to monotonicity up to n^(1/2−k)factor requires 2^{n^k} nonadaptive queries, thereby ruling out a poly(n, 1/α)query nonadaptive algorithm for such approximations. This answers a question of Seshadhri (Property Testing Review, 2014) for the case of nonadaptive algorithms. Approximating the distance to a property is closely related to tolerantly testing that property. Our lower bound stands in contrast to standard (nontolerant) testing of monotonicity that can be done nonadaptively with Otilde(n/ε^2) queries.
We obtain our lower bound by proving an analogous bound for erasureresilient testers. An αerasureresilient tester for a desired property gets oracle access to a function that has at most an α fraction of values erased. The tester has to accept (with probability at least 2/3) if the erasures can be filled in to ensure that the resulting function has the property and to reject (with probability at least 2/3) if every completion of erasures results in a function that is εfar from having the property. Our method yields the same lower bounds for unateness and being a kjunta. These lower bounds improve exponentially on the existing lower bounds for these properties.
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 Award ID(s):
 1909612
 NSFPAR ID:
 10185525
 Date Published:
 Journal Name:
 Proceedings of the 2020 ACMSIAM Symposium on Discrete Algorithms, SODA 2020
 Page Range / eLocation ID:
 19952009
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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