Generalized Uniformity Testing

TL;DR

This paper investigates the problem of testing whether an unknown discrete distribution is uniform over its support, providing new bounds and an adaptive algorithm that differ from classical known-domain uniformity testing.

Contribution

It introduces the generalized uniformity testing problem, establishes nearly tight bounds, and presents an adaptive testing algorithm for unknown supports.

Findings

Sample complexity differs significantly from known-domain case.

Nearly tight upper and lower bounds are established.

The proposed algorithm is intrinsically adaptive.

Abstract

In this work, we revisit the problem of uniformity testing of discrete probability distributions. A fundamental problem in distribution testing, testing uniformity over a known domain has been addressed over a significant line of works, and is by now fully understood. The complexity of deciding whether an unknown distribution is uniform over its unknown (and arbitrary) support, however, is much less clear. Yet, this task arises as soon as no prior knowledge on the domain is available, or whenever the samples originate from an unknown and unstructured universe. In this work, we introduce and study this generalized uniformity testing question, and establish nearly tight upper and lower bound showing that -- quite surprisingly -- its sample complexity significantly differs from the known-domain case. Moreover, our algorithm is intrinsically adaptive, in contrast to the overwhelming majority of known distribution testing algorithms.