Replicable Uniformity Testing
This addresses the issue of non-replicable algorithms in scientific studies, which can lead to contradictory results and erode public trust, by introducing a replicable method for a fundamental distribution testing problem.
The paper tackles the problem of uniformity testing with a focus on algorithmic replicability, ensuring consistent results across repeated runs, and achieves a replicable tester with sample complexity of ˜O(√n ε^{-2} ρ^{-1}), improving the typical ρ^{-2} overhead to nearly linear dependence on ρ. It also provides a nearly matching lower bound for symmetric algorithms.
Uniformity testing is arguably one of the most fundamental distribution testing problems. Given sample access to an unknown distribution $\mathbf{p}$ on $[n]$, one must decide if $\mathbf{p}$ is uniform or $\varepsilon$-far from uniform (in total variation distance). A long line of work established that uniformity testing has sample complexity $Θ(\sqrt{n}\varepsilon^{-2})$. However, when the input distribution is neither uniform nor far from uniform, known algorithms may have highly non-replicable behavior. Consequently, if these algorithms are applied in scientific studies, they may lead to contradictory results that erode public trust in science. In this work, we revisit uniformity testing under the framework of algorithmic replicability [STOC '22], requiring the algorithm to be replicable under arbitrary distributions. While replicability typically incurs a $ρ^{-2}$ factor overhead in sample complexity, we obtain a replicable uniformity tester using only $\tilde{O}(\sqrt{n} \varepsilon^{-2} ρ^{-1})$ samples. To our knowledge, this is the first replicable learning algorithm with (nearly) linear dependence on $ρ$. Lastly, we consider a class of ``symmetric" algorithms [FOCS '00] whose outputs are invariant under relabeling of the domain $[n]$, which includes all existing uniformity testers (including ours). For this natural class of algorithms, we prove a nearly matching sample complexity lower bound for replicable uniformity testing.