🤖 AI Summary
This study investigates the learnability and query efficiency gap between fixed (non-adaptive) and interactive (adaptive) testing in hypothesis testing over a finite outcome space. Under the conditional sampling model, it establishes the first necessary and sufficient condition for two distribution classes to be reliably distinguishable: a positive separation in their conditional probabilities. By measuring simulation accuracy via total variation distance and combining randomized non-adaptive designs with information-theoretic lower bounds, the work demonstrates that adaptive strategies confer at most a quadratic—rather than exponential—query advantage. Furthermore, it constructs a method that $\rho$-approximates any $T$-step adaptive strategy using only $O(N^2(T + \log(1/\rho)))$ pre-specified queries, thereby proving that the worst-case query complexity gap between adaptive and non-adaptive approaches is $\Theta(N^2)$.
📝 Abstract
Model evaluations may fix all tests before observing any responses or select later tests using earlier responses. We study this choice in a conditional-query model on a finite outcome space $\mathcal{X}$ with $|\mathcal{X}|=N$. We first ask which pairs of distribution classes can be reliably distinguished. We then ask how many additional queries are required to match an adaptive tester when all queried events must be fixed in advance. We show that learnability holds if and only if the two classes have positive separation in their pairwise conditional probabilities. When this separation is zero, the optimal worst-case error is exactly $1/2$ at every finite query budget. For any $T$-query adaptive policy and any $ρ\in (0,1)$, we construct a randomized non-adaptive procedure using $O(N^2(T + \log(1/ρ)))$ pair queries chosen before any response is observed. Its simulated transcript is within $ρ$ in total variation of the adaptive transcript, uniformly over all distributions in the model. We also construct a matching family with constant adaptive query complexity and $Ω_\varepsilon(N^2)$ non-adaptive query complexity. Consequently, the worst-case fixed-error adaptivity gap is $Θ_\varepsilon(N^2)$. Thus interaction can reduce the required number of tests by a quadratic factor, but the apparent exponential branching of an interactive evaluation does not yield an exponential query advantage.