An Automatic Inequality Prover and Instance

We consider the problem of verifying the identity of a distribution: Given the 4 description of a distribution over a discrete finite or countably infinite support, p = (p1, p2, . . .), how 5 many samples (independent draws) must one obtain from an unknown distribution, q, to distinguish, 6 with high probability, the case that p = q from the case that the total variation distance (L1 distance) 7 ‖p−q‖1 ≥ ? We resolve this question, up to constant factors, on an instance by instance basis: there 8 exist universal constants c, c′ and a function f(p, ) on the known distribution p and error parameter 9 , such that our tester distinguishes p = q from ‖p − q‖1 ≥ using f(p, ) samples with success 10 probability > 2/3, but no tester can distinguish p = q from ‖p − q‖1 ≥ c · when given c′ · f(p, ) 11 samples. The function f(p, ) is upper-bounded by a multiple of ‖p‖2/3 2 , but is more complicated. 12 This result generalizes and tightens previous results: since distributions of support at most n have 13 L2/3 norm bounded by √ n, this result immediately shows that for such distributions, O( √ n/ 2) 14 samples suffice, tightening the previous bound of O( √ n polylogn 4 ) and matching the (tight) results 15 for the case that p is the uniform distribution of support n. 16 The analysis of our very simple testing algorithm involves several hairy inequalities. To facilitate this analysis, we give a complete characterization of a general class of inequalities—generalizing Cauchy-Schwarz, Hölder’s inequality, and the monotonicity of Lp norms. Specifically, we characterize the set of sequences of triples (a, b, c)i = (a1, b1, c1), . . . , (ar, br, cr) for which it holds that for all finite sequences of positive numbers (x)j = x1, . . . and (y)j = y1, . . . ,