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# Learning and testing junta distributions with sub cube conditioning

*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.