## Abstract

Let H be a class of boolean functions and consider a composed class H^{0} that is derived from H using some arbitrary aggregation rule (for example, H^{0} may be the class of all 3-wise majority-votes of functions in H). We upper bound the Littlestone dimension of H^{0} in terms of that of H. As a corollary, we derive closure properties for online learning and private PAC learning. The derived bounds on the Littlestone dimension exhibit an undesirable exponential dependence. For private learning, we prove close to optimal bounds that circumvents this suboptimal dependency. The improved bounds on the sample complexity of private learning are derived algorithmically via transforming a private learner for the original class H to a private learner for the composed class H^{0}. Using the same ideas we show that any (proper or improper) private algorithm that learns a class of functions H in the realizable case (i.e., when the examples are labeled by some function in the class) can be transformed to a private algorithm that learns the class H in the agnostic case.

Original language | English |
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Pages (from-to) | 119-152 |

Number of pages | 34 |

Journal | Proceedings of Machine Learning Research |

Volume | 125 |

State | Published - 2020 |

Event | 33rd Conference on Learning Theory, COLT 2020 - Virtual, Online, Austria Duration: 9 Jul 2020 → 12 Jul 2020 |

### Funding

Funders | Funder number |
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National Science Foundation | DMS-1855464 |

Simons Foundation | 152/17, 1565387 |

Bloom's Syndrome Foundation | 2018267 |

Iowa Science Foundation | 281/17 |

Engineering Research Centers | 742754 |