Anaconda: A Non-Adaptive Conditional Sampling Algorithm for Distribution Testing

We investigate distribution testing with access to non-adaptive conditional samples. In the conditional sampling model, the algorithm is given the following access to a distribution: it submits a query set $S$ to an oracle, which returns a sample from the distribution conditioned on being from $S$. In the non-adaptive setting, all query sets must be specified in advance of viewing the outcomes. Our main result is the first polylogarithmic-query algorithm for equivalence testing, deciding whether two unknown distributions are equal to or far from each other. This is an exponential improvement over the previous best upper bound, and demonstrates that the complexity of the problem in this model is intermediate to the the complexity of the problem in the standard sampling model and the adaptive conditional sampling model. We also significantly improve the sample complexity for the easier problems of uniformity and identity testing. For the former, our algorithm requires only $\tilde O(\log n)$ queries, matching the information-theoretic lower bound up to a $O(\log \log n)$-factor. Our algorithm works by reducing the problem from $\ell_1$-testing to $\ell_\infty$-testing, which enjoys a much cheaper sample complexity. Necessitated by the limited power of the non-adaptive model, our algorithm is very simple to state. However, there are significant challenges in the analysis, due to the complex structure of how two arbitrary distributions may differ.