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| | {{Header | | {{Header |
| | + | |title=Distinguishing Distributions with Conditional Samples |
| | |source=bertinoro14 | | |source=bertinoro14 |
| | |who=Eldar Fischer | | |who=Eldar Fischer |
| | }} | | }} |
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| − | Suppose we are given access to two distributions $P$ and $Q$ over $\{1,2, \ldots, n\}$ and wish to test if they are the same or are at least $\epsilon$ apart under the $\ell_1$ distance. Assume that we have access to ''conditional samples'': a query consists of a set $S \subseteq \{1,2, \ldots, n\}$ and the output is a sample drawn from the conditional distribution on $S$ {{cite|ChakrabortyFGM-13|CanonneRS-14}}. In other words, if $p_j$ is the probability of drawing an element $j$ from $P$, a conditional sample from $P$ restricted to $S$ is drawn from the distribution where
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| − | $$ \text{Pr}(j) = \begin{cases} \frac{p_j}{\sum_{i \in S} p_i} & \mbox{if }j \in S, \\ 0 & \mbox{otherwise.}\end{cases} $$
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| − | It is known that if one of the distributions is fixed, then the testing problem requires at most $\tilde O(1/\epsilon^4)$ queries, which is independent of $n$ {{cite|CanonneRS-14}}.
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| − | What can we say if both distributions are unknown? The best known upper bound is $\tilde O\left( \frac{\log^5 n}{\epsilon^4} \right)$ {{cite|CanonneRS-14}}.
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| − | ==Updates==
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| − | Acharya, Canonne, and Kamath {{cite|AcharyaCK-14}} showed that $\Omega(\sqrt{\log \log n})$ conditional queries are needed in this case for some constant $\epsilon > 0$. Contrary to the case of only one distribution unknown, if both distributions are unknown, the required number of queries is a function of $n$. Falahatgar, Jafarpour, Orlitsky, Pichapathi, and Suresh {{cite|FalahatgarJOPS-15}} showed that $O\left(\frac{\log{\log{n}}}{\epsilon^5}\right)$ queries are sufficient. This determines the query complexity of the problem up to a factor of $\sqrt{\log \log n}$.
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| − | In the non-adaptive model, Kamath and Tzamos {{cite|KamathT-19}} showed that $\operatorname{poly} \log n$ conditional queries are sufficient for equivalence testing. A lower bound of $\Omega(\log n)$ by Acharya, Canonne, and Kamath {{cite|AcharyaCK-14}} for uniformity testing shows that identity and equivalence testing have complexities related by polynomial factors in the non-adaptive model, compared to the gap in the adaptive model.
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