The last-step minimax algorithm

Eiji Takimoto, Manfred K. Warmuth

Research output: Chapter in Book/Report/Conference proceedingConference contribution

15 Citations (Scopus)


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(xtt) 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.

Original languageEnglish
Title of host publicationAlgorithmic Learning Theory - 11th International Conference, ALT 2000, Proceedings
EditorsHiroki Arimura, Sanjay Jain, Arun Sharma
PublisherSpringer Verlag
Number of pages12
ISBN (Print)9783540412373
Publication statusPublished - 2000
Externally publishedYes
Event11th International Conference on Algorithmic Learning Theory, ALT 2000 - Sydney, Australia
Duration: Dec 11 2000Dec 13 2000

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Other11th International Conference on Algorithmic Learning Theory, ALT 2000

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)


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