An Automatic Inequality Prover and Instance Optimal Identity Testing

We consider the problem of verifying the identity of a distribution: Given the description of a distribution over a discrete support p = (p₁, p₂, ... , p_n), how many samples (independent draws) must one obtain from an unknown distribution, q, to distinguish, with high probability, the case that p = q from the case that the total variation distance (L₁ distance) ||p - q||1≥ ϵ? We resolve this question, up to constant factors, on an instance by instance basis: there exist universal constants c, c´ and a function f(p, ϵ) on distributions and error parameters, such that our tester distinguishes p = q from ||p-q||1≥ ϵ using f(p, ϵ) samples with success probability > 2/3, but no tester can distinguish p = q from ||p - q||1≥ c · ϵ when given c´ · f(p, ϵ) samples. The function f(p, ϵ) is upperbounded by a multiple of ||p||2/3/ϵ², but is more complicated, and is significantly smaller in some cases when p has many small domain elements, or a single large one. This result significantly generalizes and tightens previous results: since distributions of support at most n have L_2/3 norm bounded by √n, this result immediately shows that for such distributions, O(√n/ϵ²) samples suffice, tightening the previous bound of O(√npolylog/n⁴) for this class of distributions, and matching the (tight) known results for the case that p is the uniform distribution over support n. 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, Holder´s inequality, and the monotonicity of L_p norms. Specifically, we characterize the set of sequences (a)_i = a₁, . . . , ar, (b)i = b_{1<- sub>, . . . , br, (c)i = c1, ... , cr, for which it holds that for all finite sequences of positive numbers (x)j = x1,... and (y)j = y1,...,Πi=1^r (Σjx^ajⁱyi^bi)^ci≥1. For example, the standard Cauchy-Schwarz inequality corresponds to the sequences a = (1, 0, 1/2), b = (0,1, 1/2), c = (1/2 , 1/2 , -1). Our characterization is of a non-traditional nature in that it uses linear programming to compute a derivation that may otherwise have to be sought throu.gh trial and error, by hand. We do not believe such a characterization has appeared in the literature, and hope its computational nature will be useful to others, and facilitate analyses like the one here.}