Testing shape restriction properties of probability distributions in a unified way

Abstract

We study the question of testing structured properties of discrete distributions. Specifically, given sample access to an arbitrary distribution D over [n] and a property P, the goal is to distinguish between ... Building on a result of [9], we develop a general algorithm for this question, which applies to a large range of "shape-constrained" properties, including monotone, log-concave, t-modal and Poisson Binomial distributions. Our generic property tester works for properties that exclusively contain distributions which can be well approximated by L-histograms for a small (usually logarithmic in the domain size) value of L. The sample complexity of this generic approach is ... Moreover, for all cases considered, our algorithm has near-optimal sample complexity. Finally, we also describe a generic method to prove lower bounds for this problem, and use it to derive strong converses to our algorithmic results. More specifically, we use the following reduction technique: we compose the property tester for a class-C of distributions with an agnostic learner for that same class to get a tester for subset CHARD ... C for which a lower bound is known.