## Abstract

The problem of learning a binary relation between two sets of objects or between a set and itself is studied. This paper represents a binary relation between a set of size n and a set of size m as an n×m matrix of bits whose (i, j) entry is 1 if and only if the relation holds between the corresponding elements of the two sets. Polynomial prediction algorithms are presented for learning binary relations in an extended on-line learning model, where the examples are drawn by the learner, by a helpful teacher, by an adversary, or according to a uniform probability distribution on the instance space. The first part of this paper presents results for the case in which the matrix of the relation has at most k row types. It presents upper and lower bounds on the number of prediction mistakes any prediction algorithm makes when learning such a matrix under the extended on-line learning model. Furthermore, it describes a technique that simplifies the proof of expected mistake bounds against a randomly chosen query sequence. In the second part of this paper the problem of learning a binary relation that is a total order on a set is considered. A general technique using a fully polynomial randomized approximation scheme (fpras) to implement a randomized version of the halving algorithm is described. This technique is applied to the problem of learning a total order, through the use of an fpras for counting the number of extensions of a partial order, to obtain a polynomial prediction algorithm that with high probability makes at most n lg n+(lg e)lg n mistakes when an adversary selects the query sequence. The case in which a teacher or the learner selects the query sequence is also considered.

Original language | English (US) |
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Pages (from-to) | 1006-1034 |

Number of pages | 29 |

Journal | SIAM Journal on Computing |

Volume | 22 |

Issue number | 5 |

DOIs | |

State | Published - 1993 |

## All Science Journal Classification (ASJC) codes

- Computer Science(all)
- Mathematics(all)