Sharp Bounds for Generalized Uniformity Testing

TL;DR

This paper establishes tight bounds on the sample complexity for generalized uniformity testing of discrete distributions, providing an optimal, efficient testing algorithm and matching lower bounds.

Contribution

It introduces the first optimal, computationally efficient tester for generalized uniformity testing with tight sample complexity bounds.

Findings

Sample complexity is $ heta(1/(\epsilon^{4/3}\|p ext{3} ight) + 1/(\epsilon^{2} ext{2})$.

The tester is computationally efficient and matches the information-theoretic lower bound.

Provides a comprehensive characterization of the sample complexity for generalized uniformity testing.

Abstract

We study the problem of generalized uniformity testing \cite{BC17} of a discrete probability distribution: Given samples from a probability distribution $p$ over an {\em unknown} discrete domain $\mathbf{\Omega}$, we want to distinguish, with probability at least $2/3$, between the case that $p$ is uniform on some {\em subset} of $\mathbf{\Omega}$ versus $\epsilon$-far, in total variation distance, from any such uniform distribution. We establish tight bounds on the sample complexity of generalized uniformity testing. In more detail, we present a computationally efficient tester whose sample complexity is optimal, up to constant factors, and a matching information-theoretic lower bound. Specifically, we show that the sample complexity of generalized uniformity testing is $\Theta\left(1/(\epsilon^{4/3}\|p\|_3) + 1/(\epsilon^{2} \|p\|_2) \right)$.