## Abstract

How quickly can a given class of concepts be learned from examples? It is common to measure the performance of a supervised machine learning algorithm by plotting its "learning curve", that is, the decay of the error rate as a function of the number of training examples. However, the classical theoretical framework for understanding learnability, the PAC model of Vapnik-Chervonenkis and Valiant, does not explain the behavior of learning curves: the distribution-free PAC model of learning can only bound the upper envelope of the learning curves over all possible data distributions. This does not match the practice of machine learning, where the data source is typically fixed in any given scenario, while the learner may choose the number of training examples on the basis of factors such as computational resources and desired accuracy. In this paper, we study an alternative learning model that better captures such practical aspects of machine learning, but still gives rise to a complete theory of the learnable in the spirit of the PAC model. More precisely, we consider the problem of universal learning, which aims to understand the performance of learning algorithms on every data distribution, but without requiring uniformity over the distribution. The main result of this paper is a remarkable trichotomy: there are only three possible rates of universal learning. More precisely, we show that the learning curves of any given concept class decay either at an exponential, linear, or arbitrarily slow rates. Moreover, each of these cases is completely characterized by appropriate combinatorial parameters, and we exhibit optimal learning algorithms that achieve the best possible rate in each case. For concreteness, we consider in this paper only the realizable case, though analogous results are expected to extend to more general learning scenarios.

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 | 532-541 |

Number of pages | 10 |

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

- Distribution-Dependent Learning Rates
- Online Learning
- Statistical Learning