Telling Two Distributions Apart: a Tight Characterization

We consider the problem of distinguishing between two arbitrary black-box distributions defined over the domain [n], given access to $s$ samples from both. It is known that in the worst case O(n^{2/3}) samples is both necessary and sufficient, provided that the distributions have L1 difference of at least ε. However, it is also known that in many cases fewer samples suffice. We identify a new parameter, that provides an upper bound on how many samples needed, and present an efficient algorithm that requires the number of samples independent of the domain size. Also for a large subclass of distributions we provide a lower bound, that matches our upper bound up to a poly-logarithmic factor.

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