## Abstract

How many bits of information are required to PAC learn a class of hypotheses of VC dimension d? The mathematical setting we follow is that of Bassily et al., where the value of interest is the mutual information I (S; A(S)) between the input sample S and the hypothesis outputted by the learning algorithm A. We introduce a class of functions of VC dimension d over the domain X with information complexity at least Ω ( d log log ^{|X}_{d}^{|)} bits for any consistent and proper algorithm (deterministic or random). Bassily et al. proved a similar (but quantitatively weaker) result for the case d = 1. The above result is in fact a special case of a more general phenomenon we explore. We define the notion of information complexity of a given class of functions H. Intuitively, it is the minimum amount of information that an algorithm for H must retain about its input to ensure consistency and properness. We prove a direct sum result for information complexity in this context; roughly speaking, the information complexity sums when combining several classes.

Original language | English |
---|---|

Pages (from-to) | 1547-1568 |

Number of pages | 22 |

Journal | Proceedings of Machine Learning Research |

Volume | 75 |

State | Published - 2018 |

Externally published | Yes |

Event | 31st Annual Conference on Learning Theory, COLT 2018 - Stockholm, Sweden Duration: 6 Jul 2018 → 9 Jul 2018 |

### Bibliographical note

Publisher Copyright:© 2018 I. Nachum, J. Shafer & A. Yehudayoff.

## Keywords

- Direct Sum
- Information Theory
- PAC Learning
- VC Dimension

## ASJC Scopus subject areas

- Artificial Intelligence
- Software
- Control and Systems Engineering
- Statistics and Probability