Learning and testing junta distributions with sub cube conditioning

Xi Chen, Rajesh Jayaram, Amit Levi, Erik Waingarten
Proceedings of Thirty Fourth Conference on Learning Theory, PMLR 134:1060-1113, 2021.

Abstract

We study the problems of learning and testing junta distributions on $\{-1,1\}^n$ with respect to the uniform distribution, where a distribution $p$ is a $k$-junta if its probability mass function $p(x)$ depends on a subset of at most $k$ variables. The main contribution is an algorithm for finding relevant coordinates in a $k$-junta distribution with subcube conditioning (Bhattacharyya et al 2018., Canonne et al. 2019). We give two applications: An algorithm for learning $k$-junta distributions with $\tilde{O}(k/\epsilon^2) \log n + O(2^k/\epsilon^2)$ subcube conditioning queries, and an algorithm for testing $k$-junta distributions with $\tilde{O}((k + \sqrt{n})/\epsilon^2)$ subcube conditioning queries. All our algorithms are optimal up to poly-logarithmic factors. Our results show that subcube conditioning, as a natural model for accessing high-dimensional distributions, enables significant savings in learning and testing junta distributions compared to the standard sampling model. This addresses an open question posed by Aliakbarpour et al. 2016.

Cite this Paper

BibTeX
@InProceedings{pmlr-v134-chen21b, title = {Learning and testing junta distributions with sub cube conditioning}, author = {Chen, Xi and Jayaram, Rajesh and Levi, Amit and Waingarten, Erik}, booktitle = {Proceedings of Thirty Fourth Conference on Learning Theory}, pages = {1060--1113}, year = {2021}, editor = {Belkin, Mikhail and Kpotufe, Samory}, volume = {134}, series = {Proceedings of Machine Learning Research}, month = {15--19 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v134/chen21b/chen21b.pdf}, url = {https://proceedings.mlr.press/v134/chen21b.html}, abstract = {We study the problems of learning and testing junta distributions on $\{-1,1\}^n$ with respect to the uniform distribution, where a distribution $p$ is a $k$-junta if its probability mass function $p(x)$ depends on a subset of at most $k$ variables. The main contribution is an algorithm for finding relevant coordinates in a $k$-junta distribution with subcube conditioning (Bhattacharyya et al 2018., Canonne et al. 2019). We give two applications: An algorithm for learning $k$-junta distributions with $\tilde{O}(k/\epsilon^2) \log n + O(2^k/\epsilon^2)$ subcube conditioning queries, and an algorithm for testing $k$-junta distributions with $\tilde{O}((k + \sqrt{n})/\epsilon^2)$ subcube conditioning queries. All our algorithms are optimal up to poly-logarithmic factors. Our results show that subcube conditioning, as a natural model for accessing high-dimensional distributions, enables significant savings in learning and testing junta distributions compared to the standard sampling model. This addresses an open question posed by Aliakbarpour et al. 2016.} }
Endnote
%0 Conference Paper %T Learning and testing junta distributions with sub cube conditioning %A Xi Chen %A Rajesh Jayaram %A Amit Levi %A Erik Waingarten %B Proceedings of Thirty Fourth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2021 %E Mikhail Belkin %E Samory Kpotufe %F pmlr-v134-chen21b %I PMLR %P 1060--1113 %U https://proceedings.mlr.press/v134/chen21b.html %V 134 %X We study the problems of learning and testing junta distributions on $\{-1,1\}^n$ with respect to the uniform distribution, where a distribution $p$ is a $k$-junta if its probability mass function $p(x)$ depends on a subset of at most $k$ variables. The main contribution is an algorithm for finding relevant coordinates in a $k$-junta distribution with subcube conditioning (Bhattacharyya et al 2018., Canonne et al. 2019). We give two applications: An algorithm for learning $k$-junta distributions with $\tilde{O}(k/\epsilon^2) \log n + O(2^k/\epsilon^2)$ subcube conditioning queries, and an algorithm for testing $k$-junta distributions with $\tilde{O}((k + \sqrt{n})/\epsilon^2)$ subcube conditioning queries. All our algorithms are optimal up to poly-logarithmic factors. Our results show that subcube conditioning, as a natural model for accessing high-dimensional distributions, enables significant savings in learning and testing junta distributions compared to the standard sampling model. This addresses an open question posed by Aliakbarpour et al. 2016.
APA
Chen, X., Jayaram, R., Levi, A. & Waingarten, E.. (2021). Learning and testing junta distributions with sub cube conditioning. Proceedings of Thirty Fourth Conference on Learning Theory, in Proceedings of Machine Learning Research 134:1060-1113 Available from https://proceedings.mlr.press/v134/chen21b.html.

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