Tight Bounds for Equivalence Testing with Non-Adaptive Conditional Samples

2026-10-07

AI Summary

This study addresses the problem of distribution equivalence testing under the non-adaptive conditional sampling model, aiming to determine whether two unknown distributions are identical. Methodologically, by integrating techniques from distribution testing theory and information-theoretic lower bound analysis, it systematically derives theoretical bounds on query complexity. The primary contributions are threefold: first, it proves that uniformity, identity, and equivalence testing all exhibit a query complexity of Θ̃(log n) within this model; second, it establishes that the necessary and sufficient number of queries for equivalence testing is Θ̃(log n/ε²). These findings yield the first tight bound for this problem, thereby providing a complete theoretical characterization of distribution testing under the conditional sampling framework.

Abstract

We study distribution testing with access to non-adaptive conditional samples. Specifically, we give tight bounds for equivalence testing, determining whether two unknown distributions are equal to or $\varepsilon$-far from each other in total variation distance. Our algorithm and lower bound show that $\tilde Θ\left(\frac{\log n}{\varepsilon^2}\right)$ queries are necessary and sufficient for this problem. These results demonstrate that the complexity of uniformity, identity, and equivalence testing with non-adaptive conditional samples are all $\tilde Θ(\log n)$.

Problem

• distribution testing
• equivalence testing
• non-adaptive conditional samples
• query complexity
• total variation distance

Innovation

• Equivalence Testing
• Non-Adaptive Conditional Samples
• Distribution Testing
• Query Complexity
• Total Variation Distance