An Automatic Inequality Prover and Instance Optimal Identity Testing

Gregory Valiant, Paul Valiant

2014 (modified: 17 May 2023)FOCS 2014Readers: Everyone

Abstract: We consider the problem of verifying the identity of a distribution: Given the description of a distribution over a discrete support p = (p 1 , p 2 , ..., 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 (L 1 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/ϵ 2 , 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/ϵ 2 ) samples suffice, tightening the previous bound of O(√npolylog/n 4 ) 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 = a 1 , ..., ar, (b)i = b 1 , ..., br, (c)i = c 1 , ..., cr, for which it holds that for all finite sequences of positive numbers (x) j = x 1 ,... and (y) j = y 1 ,...,Π i=1 r (Σ j x a j i y i b i ) 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.

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