Learning and Testing Junta Distributions with Subcube Conditioning

We study the problems of learning and testing junta distributions on -1,1n 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 [BC18, CCKLW20]. We give two applications: 1. An algorithm for learning k-junta distributions with tildeO(k/ε2) log n + O(2k/ε2) subcube conditioning queries, and 2. An algorithm for testing k-junta distributions with tildeO((k + √n)/ε2) 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, Blais, and Rubinfeld [ABR17].