## Abstract

Boosting is a celebrated machine learning approach which is based on the idea of combining weak and moderately inaccurate hypotheses to a strong and accurate one. We study boosting under the assumption that the weak hypotheses belong to a class of bounded capacity. This assumption is inspired by the common convention that weak hypotheses are "rules-of-thumbs"from an "easy-to-learn class". (Schapire and Freund '12, Shalev-Shwartz and Ben-David '14.) Formally, we assume the class of weak hypotheses has a bounded VC dimension. We focus on two main questions: (i) Oracle Complexity: How many weak hypotheses are needed in order to produce an accurate hypothesis? We design a novel boosting algorithm and demonstrate that it circumvents a classical lower bound by Freund and Schapire ('95, '12). Whereas the lower bound shows that ?(?2) weak hypotheses with ?-margin are sometimes necessary, our new method requires only O(?) weak hypothesis, provided that they belong to a class of bounded VC dimension. Unlike previous boosting algorithms which aggregate the weak hypotheses by majority votes, the new boosting algorithm uses more complex ("deeper") aggregation rules. We complement this result by showing that complex aggregation rules are in fact necessary to circumvent the aforementioned lower bound. (ii) Expressivity: Which tasks can be learned by boosting weak hypotheses from a bounded VC class? Can complex concepts that are "far away"from the class be learned? Towards answering the first question we identify a combinatorial-geometric parameter which captures the expressivity of base-classes in boosting. As a corollary we provide an affirmative answer to the second question for many well-studied classes, including half-spaces and decision stumps. Along the way, we establish and exploit connections with Discrepancy Theory.

Original language | English (US) |
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Title of host publication | STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing |

Editors | Samir Khuller, Virginia Vassilevska Williams |

Publisher | Association for Computing Machinery |

Pages | 481-489 |

Number of pages | 9 |

ISBN (Electronic) | 9781450380539 |

DOIs | |

State | Published - Jun 15 2021 |

Event | 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021 - Virtual, Online, Italy Duration: Jun 21 2021 → Jun 25 2021 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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ISSN (Print) | 0737-8017 |

### Conference

Conference | 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021 |
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Country/Territory | Italy |

City | Virtual, Online |

Period | 6/21/21 → 6/25/21 |

## All Science Journal Classification (ASJC) codes

- Software

## Keywords

- Boosting
- Machine Learning