Instance-Optimal Uniformity Testing and Tracking

In the uniformity testing task, an algorithm is provided with samples from an unknown probability distribution over a (known) finite domain, and must decide whether it is the uniform distribution, or, alternatively, if its total variation distance from uniform exceeds some input distance parameter. This question has received a significant amount of interest and its complexity is, by now, fully settled. Yet, we argue that it fails to capture many scenarios of interest, and that its very definition as a gap problem in terms of a prespecified distance may lead to suboptimal performance. To address these shortcomings, we introduce the problem of uniformity tracking, whereby an algorithm is required to detect deviations from uniformity (however they may manifest themselves) using as few samples as possible, and be competitive against an optimal algorithm knowing the distribution profile in hindsight. Our main contribution is a $\operatorname{polylog}(\operatorname{opt})$-competitive uniformity tracking algorithm. We obtain this result by leveraging new structural results on Poisson mixtures, which we believe to be of independent interest.