Rounds of adaptivity required for optimal equivalence testing

Determine how many rounds of adaptivity are required in conditional sampling to reduce the sample complexity of equivalence testing between two unknown distributions over [n] from the optimal non-adaptive bound \(\tilde O(\log n)\) to the optimal fully adaptive bound \(\tilde O(\log\log n)\).

Background

The paper establishes that equivalence testing with non-adaptive conditional samples requires $\tilde\Theta(\log n/\varepsilon^2)$ samples, while prior work gives an essentially optimal $\tilde O(\log\log n)$-sample algorithm in the fully adaptive conditional sampling model. For identity testing, the paper notes that one round of adaptivity already achieves the full power of adaptive conditional samples, contrasting with the unresolved situation for equivalence testing.

The open problem is to characterize the amount of adaptivity, measured by the number of rounds, needed for equivalence testing to attain the fully adaptive sample complexity rather than the non-adaptive $\tilde O(\log n)$ complexity.