## Abstract

In this paper, we study the problem of computing U-statistics of degree 2, i.e., quantities that come in the form of averages over pairs of data points, in the local model of differential privacy (LDP). The class of U-statistics covers many statistical estimates of interest, including Gini mean difference, Kendall's tau coefficient and Area under the ROC Curve (AUC), as well as empirical risk measures for machine learning problems such as ranking, clustering and metric learning. We first introduce an LDP protocol based on quantizing the data into bins and applying randomized response, which guarantees an epsilon-LDP estimate with a Mean Squared Error (MSE) of O(1/root n epsilon) under regularity assumptions on the U-statistic or the data distribution. We then propose a specialized protocol for AUC based on a novel use of hierarchical histograms that achieves MSE of O (alpha(3)/n epsilon(2)) for arbitrary data distribution. We also show that 2-party secure computation allows to design a protocol with MSE of O(1/n epsilon(2)), without any assumption on the kernel function or data distribution and with total communication linear in the number of users n. Finally, we evaluate the performance of our protocols through experiments on synthetic and real datasets.

Original language | English |
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Title of host publication | INTERNATIONAL CONFERENCE ON ARTIFICIAL INTELLIGENCE AND STATISTICS, VOL 108 |

Editors | S Chiappa, R Calandra |

Publisher | ADDISON-WESLEY |

Pages | 1573-1582 |

Number of pages | 10 |

Publication status | Published - 2020 |

MoE publication type | A4 Article in a conference publication |

Event | International Conference on Artificial Intelligence and Statistics - Palermo, Italy Duration: 3 Jun 2020 → 5 Jun 2020 Conference number: 23 |

### Publication series

Name | Proceedings of Machine Learning Research |
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Publisher | ADDISON-WESLEY PUBL CO |

Volume | 108 |

ISSN (Print) | 2640-3498 |

### Conference

Conference | International Conference on Artificial Intelligence and Statistics |
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Abbreviated title | AISTATS |

Country | Italy |

City | Palermo |

Period | 03/06/2020 → 05/06/2020 |