### Abstract

We describe online algorithms for learning a rotation from pairs of unit vectors in ℝ^{n}. We show that the expected regret of our online algorithm compared to the best fixed rotation chosen offline is O(√nL), where L is the loss of the best rotation. We also give a lower bound that proves that this expected regret bound is optimal within a constant factor. This resolves an open problem posed in COLT 2008. Our online algorithm for choosing a rotation matrix in each trial is based on the Follow-The-Perturbed-Leader paradigm. It adds a random spectral perturbation to the matrix characterizing the loss incurred so far and then chooses the best rotation matrix for that loss. We also show that any deterministic algorithm for learning rotations has Ω(T) regret in the worst case.

Original language | English (US) |
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Title of host publication | COLT 2010 - The 23rd Conference on Learning Theory |

Pages | 144-154 |

Number of pages | 11 |

State | Published - Dec 1 2010 |

Externally published | Yes |

Event | 23rd Conference on Learning Theory, COLT 2010 - Haifa, Israel Duration: Jun 27 2010 → Jun 29 2010 |

### Publication series

Name | COLT 2010 - The 23rd Conference on Learning Theory |
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### Other

Other | 23rd Conference on Learning Theory, COLT 2010 |
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Country | Israel |

City | Haifa |

Period | 6/27/10 → 6/29/10 |

### All Science Journal Classification (ASJC) codes

- Education

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## Cite this

*COLT 2010 - The 23rd Conference on Learning Theory*(pp. 144-154). (COLT 2010 - The 23rd Conference on Learning Theory).