Learning and Testing Junta Distributions with Subcube Conditioning

TL;DR

This paper introduces efficient algorithms for learning and testing junta distributions on high-dimensional binary spaces using subcube conditioning, significantly reducing query complexity and addressing open questions in the field.

Contribution

The paper presents new algorithms for learning and testing $k$-junta distributions with optimal or near-optimal query complexity using subcube conditioning, advancing the understanding of distribution access models.

Findings

Algorithms are optimal up to poly-logarithmic factors.

Subcube conditioning enables significant savings over standard sampling.

Addresses an open question by Aliakbarpour, Blais, and Rubinfeld.

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 [BC18, CCKLW20]. We give two applications: 1. An algorithm for learning $k$-junta distributions with $\tilde{O}(k/\epsilon^2) \log n + O(2^k/\epsilon^2)$ subcube conditioning queries, and 2. 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, Blais, and Rubinfeld [ABR17].