### Abstract

We study the tradeoff between the statistical error and communication cost of distributed statistical estimation problems in high dimensions. In the distributed sparse Gaussian mean estimation problem, each of the m machines receives n data points from a d-dimensional Gaussian distribution with unknown mean θ which is promised to be k-sparse. The machines communicate by message passing and aim to estimate the mean θ We provide a tight (up to logarithmic factors) tradeoff between the estimation error and the number of bits communicated between the machines. This directly leads to a lower bound for the distributed sparse linear regression problem: to achieve the statistical minimax error, the total communication is at least?(min{n,d}m), where n is the number of observations that each machine receives and d is the ambient dimension. These lower bound results improve upon Shamir (NIPS'14) and Steinhardt, Duchi (COLT'15) by allowing a multi-round interactive communication model. We also give the first optimal simultaneous protocol in the dense case for mean estimation. As our main technique, we prove a distributed data processing inequality, as a generalization of usual data processing inequalities, which might be of independent interest and useful for other problems.

Original language | English (US) |
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Title of host publication | STOC 2016 - Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing |

Editors | Yishay Mansour, Daniel Wichs |

Publisher | Association for Computing Machinery |

Pages | 1011-1020 |

Number of pages | 10 |

ISBN (Electronic) | 9781450341325 |

DOIs | |

State | Published - Jun 19 2016 |

Event | 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016 - Cambridge, United States Duration: Jun 19 2016 → Jun 21 2016 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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Volume | 19-21-June-2016 |

ISSN (Print) | 0737-8017 |

### Other

Other | 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016 |
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Country | United States |

City | Cambridge |

Period | 6/19/16 → 6/21/16 |

### All Science Journal Classification (ASJC) codes

- Software

### Keywords

- Communication complexity
- Information complexity
- Statistical estimation

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## Cite this

*STOC 2016 - Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing*(pp. 1011-1020). (Proceedings of the Annual ACM Symposium on Theory of Computing; Vol. 19-21-June-2016). Association for Computing Machinery. https://doi.org/10.1145/2897518.2897582