Redundancy capacity theorem for on-line learning under a certain form of hypotheses class

In this paper we consider the problem of online learning in the stochastic setting under a certain form of hypotheses class. We prove an equivalence between the minimax redundancy and capacity of the channel between the class parameters and the labels conditioned on the data features (side information). Our proof extends Gallager's Redundancy Capacity theorem for universal prediction to on-line learning with the considered form of hypotheses class. Moreover, this result confirms the optimality of previous ad-hoc universal learners, or universal predictors with side information, but more importantly, extends these previous results to more general hypotheses classes.