### Abstract

We consider on-line density estimation with a parameterized density from an exponential family. In each trial t the learner predicts a parameter θ_{t}. Then it receives an instance x_{t} chosen by the adversary and incurs loss -ln p(x_{t}|θ_{t}) which is the negative log-likelihood of x_{t} w.r.t. the predicted density of the learner. The performance of the learner is measured by the regret defined as the total loss of the learner minus the total loss of the best parameter chosen off-line. We develop an algorithm called the Last-step Minimax Algorithm that predicts with the minimax optimal parameter assuming that the current trial is the last one. For one-dimensional exponential families, we give an explicit form of the prediction of the Last-step Minimax Algorithm and show that its regret is O(ln T), where T is the number of trials. In particular, for Bernoulli density estimation the Last-step Minimax Algorithm is slightly better than the standard Krichevsky-Trofimov probability estimator.

Original language | English |
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Title of host publication | Algorithmic Learning Theory - 11th International Conference, ALT 2000, Proceedings |

Editors | Sanjay Jain, Hiroki Arimura, Arun Sharma |

Publisher | Springer Verlag |

Pages | 279-290 |

Number of pages | 12 |

ISBN (Print) | 9783540412373 |

Publication status | Published - Jan 1 2000 |

Externally published | Yes |

Event | 11th International Conference on Algorithmic Learning Theory, ALT 2000 - Sydney, Australia Duration: Dec 11 2000 → Dec 13 2000 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 1968 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 11th International Conference on Algorithmic Learning Theory, ALT 2000 |
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Country | Australia |

City | Sydney |

Period | 12/11/00 → 12/13/00 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Algorithmic Learning Theory - 11th International Conference, ALT 2000, Proceedings*(pp. 279-290). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1968). Springer Verlag.

**The last-step minimax algorithm.** / Takimoto, Eiji; Warmuth, Manfred K.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Algorithmic Learning Theory - 11th International Conference, ALT 2000, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 1968, Springer Verlag, pp. 279-290, 11th International Conference on Algorithmic Learning Theory, ALT 2000, Sydney, Australia, 12/11/00.

}

TY - GEN

T1 - The last-step minimax algorithm

AU - Takimoto, Eiji

AU - Warmuth, Manfred K.

PY - 2000/1/1

Y1 - 2000/1/1

N2 - We consider on-line density estimation with a parameterized density from an exponential family. In each trial t the learner predicts a parameter θt. Then it receives an instance xt chosen by the adversary and incurs loss -ln p(xt|θt) which is the negative log-likelihood of xt w.r.t. the predicted density of the learner. The performance of the learner is measured by the regret defined as the total loss of the learner minus the total loss of the best parameter chosen off-line. We develop an algorithm called the Last-step Minimax Algorithm that predicts with the minimax optimal parameter assuming that the current trial is the last one. For one-dimensional exponential families, we give an explicit form of the prediction of the Last-step Minimax Algorithm and show that its regret is O(ln T), where T is the number of trials. In particular, for Bernoulli density estimation the Last-step Minimax Algorithm is slightly better than the standard Krichevsky-Trofimov probability estimator.

AB - We consider on-line density estimation with a parameterized density from an exponential family. In each trial t the learner predicts a parameter θt. Then it receives an instance xt chosen by the adversary and incurs loss -ln p(xt|θt) which is the negative log-likelihood of xt w.r.t. the predicted density of the learner. The performance of the learner is measured by the regret defined as the total loss of the learner minus the total loss of the best parameter chosen off-line. We develop an algorithm called the Last-step Minimax Algorithm that predicts with the minimax optimal parameter assuming that the current trial is the last one. For one-dimensional exponential families, we give an explicit form of the prediction of the Last-step Minimax Algorithm and show that its regret is O(ln T), where T is the number of trials. In particular, for Bernoulli density estimation the Last-step Minimax Algorithm is slightly better than the standard Krichevsky-Trofimov probability estimator.

UR - http://www.scopus.com/inward/record.url?scp=84974698795&partnerID=8YFLogxK

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M3 - Conference contribution

AN - SCOPUS:84974698795

SN - 9783540412373

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 279

EP - 290

BT - Algorithmic Learning Theory - 11th International Conference, ALT 2000, Proceedings

A2 - Jain, Sanjay

A2 - Arimura, Hiroki

A2 - Sharma, Arun

PB - Springer Verlag

ER -