Tolerant Testing of High-Dimensional Samplers with Subcube Conditioning
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We study the tolerant testing problem for high-dimensional samplers. Given as input two samplers 𝒫 and 𝒬 over the n-dimensional space {0,1}^n, and two parameters ε_2 > ε_1, the goal of tolerant testing is to test whether the distributions generated by 𝒫 and 𝒬 are ε_1-close or ε_2-far. Since exponential lower bounds (in n) are known for the problem in the standard sampling model, research has focused on models where one can draw conditional samples. Among these models, subcube conditioning (𝖲𝖴𝖡𝖢𝖮𝖭𝖣), which allows conditioning on arbitrary subcubes of the domain, holds the promise of widespread adoption in practice owing to its ability to capture the natural behavior of samplers in constrained domains. To translate the promise into practice, we need to overcome two crucial roadblocks for tests based on 𝖲𝖴𝖡𝖢𝖮𝖭𝖣: the prohibitively large number of queries (𝒪̃(n^5/ε_2^5)) and limitation to non-tolerant testing (i.e., ε_1 = 0). The primary contribution of this work is to overcome the above challenges: we design a new tolerant testing methodology (i.e., ε_1 ≥ 0) that allows us to significantly improve the upper bound to 𝒪̃(n^3/(ε_2-ε_1)^5).
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