A New Approach for Testing Properties of Discrete Distributions

TL;DR

This paper introduces a unified, modular approach for distribution testing that achieves sample-optimal algorithms for a wide range of problems, simplifying analysis and extending to new metrics.

Contribution

It presents a novel reduction-based framework that yields the first sample-optimal testers for many distribution testing problems and simplifies existing estimators.

Findings

Achieves sample-optimal testers for identity, closeness, independence, and histogram testing.

Provides the first nearly instance-optimal algorithm for testing distribution equivalence.

Introduces a new information-theoretic lower bound technique for distribution property testing.

Abstract

In this work, we give a novel general approach for distribution testing. We describe two techniques: our first technique gives sample-optimal testers, while our second technique gives matching sample lower bounds. As a consequence, we resolve the sample complexity of a wide variety of testing problems. Our upper bounds are obtained via a modular reduction-based approach. Our approach yields optimal testers for numerous problems by using a standard $\ell_2$-identity tester as a black-box. Using this recipe, we obtain simple estimators for a wide range of problems, encompassing most problems previously studied in the TCS literature, namely: (1) identity testing to a fixed distribution, (2) closeness testing between two unknown distributions (with equal/unequal sample sizes), (3) independence testing (in any number of dimensions), (4) closeness testing for collections of distributions, and (5) testing histograms. For all of these problems, our testers are sample-optimal, up to constant factors. With the exception of (1), ours are the {\em first sample-optimal testers for the corresponding problems.} Moreover, our estimators are significantly simpler to state and analyze compared to previous results. As an application of our reduction-based technique, we obtain the first {\em nearly instance-optimal} algorithm for testing equivalence between two {\em unknown} distributions. Moreover, our technique naturally generalizes to other metrics beyond the $\ell_1$-distance. Our lower bounds are obtained via a direct information-theoretic approach: Given a candidate hard instance, our proof proceeds by bounding the mutual information between appropriate random variables. While this is a classical method in information theory, prior to our work, it had not been used in distribution property testing.