Online learning with prior knowledge

The standard so-called experts algorithms are methods for utilizing a given set of "experts" to make good choices in a sequential decision-making problem. In the standard setting of experts algorithms, the decision maker chooses repeatedly in the same "state" based on information about how the different experts would have performed if chosen to be followed. In this paper we seek to extend this framework by introducing state information. More precisely, we extend the framework by allowing an experts algorithm to rely on state information, namely, partial information about the cost function, which is revealed to the decision maker before the latter chooses an action. This extension is very natural in prediction problems. For illustration, an experts algorithm, which is supposed to predict whether the next day will be rainy, can be extended to predicting the same given the current temperature. We introduce new algorithms, which attain optimal performance in the new framework, and apply to more general settings than variants of regression that have been considered in the statistics literature.