Sharp Bounds for Generalized Uniformity Testing

We study the problem of generalized uniformity testing [BC17] of a discrete probability distribution: Given samples from a probability distribution p over an unknown discrete domain Ω, we want to distinguish, with probability at least 2/3, between the case that p is uniform on some subset of Ω versus -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 Θ ( 1/( ‖p‖3) + 1/( ‖p‖2) ) .