Agnostic Online learnability

We study a fundamental question. What classes of hypotheses are learnable in the online learning model? The analogous question in the PAC learning model [12] was addressed by Vapnik and others [13], who showed that the VC dimension characterizes the learnability of a hypothesis class. In his influential work, Littlestone [9] studied the online learnability of hypothesis classes, but only in the realizable case, namely, assuming that there exists a hypothesis in the class that perfectly explains the entire data. In this paper we study the online learnability in the agnostic case, namely, no hypothesis perfectly predicts the entire data, and our goal is to minimize regret. We first present an impossibility result, discovered by Cover in the context of universal prediction of individual sequences, which implies that even a class whose Littlestone’s dimension is only 1, is not learnable in the agnostic online learning model. We then overcome the impossibility result by allowing randomized predictions, and show that in this case Littlestone’s dimension does capture the learnability of hypotheses classes in the agnostic online learning model.