## Abstract

We consider the problem of learning distributions in the presence of irrelevant features. This problem is formalized by introducing a new notion of k-junta distributions. Informally, a distribution D over the domain X^{n} is a k-junta distribution with respect to another distribution U over the same domain if there is a set J ⊆ [n] of size |J| ≤ k that captures the difference between D and U. We show that it is possible to learn k-junta distributions with respect to the uniform distribution over the Boolean hypercube {0, 1}^{n} in time poly(n^{k}, 1/ε). This result is obtained via a new Fourier-based learning algorithm inspired by the Low-Degree Algorithm of Linial, Mansour, and Nisan (1993). We also consider the problem of testing whether an unknown distribution is a k-junta distribution with respect to the uniform distribution. We give a nearly-optimal algorithm for this task. Both the analysis of the algorithm and the lower bound showing its optimality are obtained by establishing connections between the problem of testing junta distributions and testing uniformity of weighted collections of distributions.

Original language | English |
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Pages (from-to) | 19-46 |

Number of pages | 28 |

Journal | Journal of Machine Learning Research |

Volume | 49 |

Issue number | June |

State | Published - 6 Jun 2016 |

Event | 29th Conference on Learning Theory, COLT 2016 - New York, United States Duration: 23 Jun 2016 → 26 Jun 2016 |

### Funding

Funders | Funder number |
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National Science Foundation | CCF-1420692, CCF-1065125 |

Iowa Science Foundation | 1536/14 |

Natural Sciences and Engineering Research Council of Canada |

## Keywords

- Fourier analysis
- Juntas
- Learning distributions
- Property testing