TR16-074 Authors: Ilias Diakonikolas, Daniel Kane

Publication: 9th May 2016 11:10

Downloads: 1295

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We study problems in distribution property testing:

Given sample access to one or more unknown discrete distributions,

we want to determine whether they have some global property or are $\epsilon$-far

from having the property in $\ell_1$ distance (equivalently, total variation distance, or ``statistical distance'').

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 important application of our reduction-based technique,

we obtain the first {\em nearly instance-optimal} algorithm for testing equivalence between

two {\em unknown} distributions. The sample complexity of our algorithm

depends on the {\em structure of the unknown distributions} -- as opposed to merely their domain size --

and is much better compared to the worst-case optimal $\ell_1$-tester in most natural instances.

Moreover, our technique naturally generalizes to other metrics beyond the $\ell_1$-distance.

As an illustration of its flexibility, we use it to obtain the first near-optimal equivalence tester

under the Hellinger 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.

Previous lower bounds relied either on the birthday paradox, or

on moment-matching and were thus restricted to symmetric properties.

Our lower bound approach does not suffer from any such restrictions

and gives tight sample lower bounds for the aforementioned problems.