TY - GEN

T1 - Monotone probability distributions over the Boolean cube can be learned with sublinear samples

AU - Rubinfeld, Ronitt

AU - Vasilyan, Arsen

N1 - Publisher Copyright:
© Ronitt Rubinfeld and Arsen Vasilyan.

PY - 2020/1

Y1 - 2020/1

N2 - A probability distribution over the Boolean cube is monotone if flipping the value of a coordinate from zero to one can only increase the probability of an element. Given samples of an unknown monotone distribution over the Boolean cube, we give (to our knowledge) the first algorithm that learns an approximation of the distribution in statistical distance using a number of samples that is sublinear in the domain. To do this, we develop a structural lemma describing monotone probability distributions. The structural lemma has further implications to the sample complexity of basic testing tasks for analyzing monotone probability distributions over the Boolean cube: We use it to give nontrivial upper bounds on the tasks of estimating the distance of a monotone distribution to uniform and of estimating the support size of a monotone distribution. In the setting of monotone probability distributions over the Boolean cube, our algorithms are the first to have sample complexity lower than known lower bounds for the same testing tasks on arbitrary (not necessarily monotone) probability distributions. One further consequence of our learning algorithm is an improved sample complexity for the task of testing whether a distribution on the Boolean cube is monotone.

AB - A probability distribution over the Boolean cube is monotone if flipping the value of a coordinate from zero to one can only increase the probability of an element. Given samples of an unknown monotone distribution over the Boolean cube, we give (to our knowledge) the first algorithm that learns an approximation of the distribution in statistical distance using a number of samples that is sublinear in the domain. To do this, we develop a structural lemma describing monotone probability distributions. The structural lemma has further implications to the sample complexity of basic testing tasks for analyzing monotone probability distributions over the Boolean cube: We use it to give nontrivial upper bounds on the tasks of estimating the distance of a monotone distribution to uniform and of estimating the support size of a monotone distribution. In the setting of monotone probability distributions over the Boolean cube, our algorithms are the first to have sample complexity lower than known lower bounds for the same testing tasks on arbitrary (not necessarily monotone) probability distributions. One further consequence of our learning algorithm is an improved sample complexity for the task of testing whether a distribution on the Boolean cube is monotone.

KW - Estimating support size

KW - Learning distributions

KW - Monotone probability distributions

UR - http://www.scopus.com/inward/record.url?scp=85077995837&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ITCS.2020.28

DO - 10.4230/LIPIcs.ITCS.2020.28

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AN - SCOPUS:85077995837

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 11th Innovations in Theoretical Computer Science Conference, ITCS 2020

A2 - Vidick, Thomas

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 11th Innovations in Theoretical Computer Science Conference, ITCS 2020

Y2 - 12 January 2020 through 14 January 2020

ER -