Replicable Uniformity Testing

TL;DR

This paper introduces a new replicable uniformity testing algorithm with improved sample complexity, addressing issues of non-replicability in existing methods, and establishes lower bounds for symmetric algorithms within this framework.

Contribution

The paper presents the first nearly linear sample complexity replicable uniformity tester and analyzes the fundamental limits for symmetric algorithms in this setting.

Findings

Developed a replicable uniformity tester with $ ilde{O}( extstylerac{ oot{2} }{ extstylerac{1}{ ho}})$ samples.

Proved a nearly matching lower bound for symmetric algorithms in replicable uniformity testing.

Addressed the non-replicability problem in classical uniformity testing algorithms.

Abstract

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 $\Theta(\sqrt{n}\varepsilon^{-2})$. 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 $\rho^{-2}$ factor overhead in sample complexity, we obtain a replicable uniformity tester using only $\tilde{O}(\sqrt{n} \varepsilon^{-2} \rho^{-1})$ samples. To our knowledge, this is the first replicable learning algorithm with (nearly) linear dependence on $\rho$. 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.