Testing Shape Restrictions of Discrete Distributions

TL;DR

This paper introduces a versatile, efficient algorithm for testing whether discrete distributions with certain shape constraints meet specified properties, achieving near-optimal sample complexity and extending to tolerant testing.

Contribution

It provides the first non-trivial testers for many shape-constrained classes and a generic method for establishing lower bounds, advancing the understanding of distribution property testing.

Findings

Near-optimal sample complexity for multiple distribution classes

First non-trivial testers for several shape-constrained distributions

Nearly tight bounds for tolerant testing

Abstract

We study the question of testing structured properties (classes) of discrete distributions. Specifically, given sample access to an arbitrary distribution $D$ over $[n]$ and a property $\mathcal{P}$, the goal is to distinguish between $D\in\mathcal{P}$ and $\ell_1(D,\mathcal{P})>\varepsilon$. We develop a general algorithm for this question, which applies to a large range of "shape-constrained" properties, including monotone, log-concave, $t$-modal, piecewise-polynomial, and Poisson Binomial distributions. Moreover, for all cases considered, our algorithm has near-optimal sample complexity with regard to the domain size and is computationally efficient. For most of these classes, we provide the first non-trivial tester in the literature. In addition, we also describe a generic method to prove lower bounds for this problem, and use it to show our upper bounds are nearly tight. Finally, we extend some of our techniques to tolerant testing, deriving nearly-tight upper and lower bounds for the corresponding questions.