Predicting a binary sequence almost as well as the optimal biased coin
COLT '96 Proceedings of the ninth annual conference on Computational learning theory
Extended Stochastic Complexity and Minimax Relative Loss Analysis
ALT '99 Proceedings of the 10th International Conference on Algorithmic Learning Theory
The Minimax Strategy for Gaussian Density Estimation. pp
COLT '00 Proceedings of the Thirteenth Annual Conference on Computational Learning Theory
Relative loss bounds for on-line density estimation with the exponential family of distributions
UAI'99 Proceedings of the Fifteenth conference on Uncertainty in artificial intelligence
Fisher information and stochastic complexity
IEEE Transactions on Information Theory
A decision-theoretic extension of stochastic complexity and its applications to learning
IEEE Transactions on Information Theory
Asymptotic minimax regret for data compression, gambling, and prediction
IEEE Transactions on Information Theory
Learning locally minimax optimal Bayesian networks
International Journal of Approximate Reasoning
Weighted last-step min-max algorithm with improved sub-logarithmic regret
ALT'12 Proceedings of the 23rd international conference on Algorithmic Learning Theory
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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.