Problem 66: Distinguishing Distributions with Conditional Samples

Suggested by	Eldar Fischer
Source	Bertinoro 2014
Short link	https://sublinear.info/66

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 $S$ [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 $$ \text{Pr}(j) = \begin{cases} \frac{p_j}{\sum_{i \in S} p_i} & \mbox{if }j \in S, \\ 0 & \mbox{otherwise.}\end{cases} $$ 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 $n$ [CanonneRS-14].

What can we say if both distributions are unknown? The best known upper bound is $\tilde O\left( \frac{\log^5 n}{\epsilon^4} \right)$ [CanonneRS-14].

Updates

Acharya, Canonne, and Kamath [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 $n$. Concurrently, in 2015, Falahatgar, Jafarpour, Orlitsky, Pichapathi, and Suresh^{(citation missing)} showed that $O(\frac{\log{\log{n}}}{\varepsilon^5})$ queries are sufficient, so that the query complexity of this problem is determined up to a factor of $\sqrt{\log \log n}$.