The Last-Step Minimax Algorithm

We consider on-line density estimation with a parameterized density from an exponential family. In each trial t the learner predicts a parameter t. Then it receives an instance xt chosen by the adversary and incurs loss ln p(xtj t) 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 de ned as the total loss of the learner minus the total loss of the best parameter chosen o -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(lnT ), 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-Tro mov probability estimator.