We review and relate some classical and recent results from the theory of online learning based on discrete classes of models or predictors. Among these frameworks, Bayesian methods, MDL, and prediction (or action) with expert advice are studied. We will discuss ways to work with universal base classes corresponding to sets of all programs on some fixed universal Turing machine, resulting in universal induction schemes.

The corrected version of the Abstract should read as follows: Abstract—Minimum description length (MDL) is an important principle for induction and prediction, with strong relations to op- timal Bayesian learning. This paper deals with learning processes which are not necessarily independent and identically distributed, by means of two-part MDL, where the underlying model class is countable. We consider the online learning framework, i.e., obser- vations come in one by one, and the predictor is allowed to up- date its state of mind after each time step. We identify two ways of predicting by MDL for this setup, namely, a static and a dynamic one. (A third variant, hybrid MDL, will turn out inferior.) We will prove that under the only assumption that the data is generated by a distribution contained in the model class, the MDL predictions converge to the true values almost surely. This is accomplished by proving finite bounds on the quadratic, the Hellinger, and the KullbackLeibler loss of the MDL learner, which are, however, ex- ponentially worse than for Bayesian prediction. We demonstrate that these bounds are sharp, even for model classes containing only Bernoulli distributions. We show how these bounds imply regret bounds for arbitrary loss functions. Our results apply to a wide range of setups, namely, sequence prediction, pattern classification, regression, and universal induction in the sense of algorithmic in- formation theory among others.

When applying aggregating strategies to Prediction with Expert Advice (PEA), the learning rate must be adaptively tuned. The natural choice of sqrt(complexity/current loss) renders the analysis of Weighted Majority (WM) derivatives quite complicated. In particular, for arbitrary weights there have been no results proven so far. The analysis of the alternative Follow the Perturbed Leader (FPL) algorithm from Kalai and Vempala (2003) based on Hannan's algorithm is easier. We derive loss bounds for adaptive learning rate and both finite expert classes with uniform weights and countable expert classes with arbitrary weights. For the former setup, our loss bounds match the best known results so far, while for the latter our results are new.

Minimum description length (MDL) is an important principle for induction and prediction, with strong relations to optimal Bayesian learning. This paper deals with learning processes which are independent and identically distributed (i.i.d.) by means of two-part MDL, where the underlying model class is countable. We consider the online learning framework, i.e., observations come in one by one, and the predictor is allowed to update its state of mind after each time step. We identify two ways of predicting by MDL for this setup, namely, a static and a dynamic one. (A third variant, hybrid MDL, will turn out inferior.) We will prove that under the only assumption that the data is generated by a distribution contained in the model class, the MDL predictions converge to the true values almost surely. This is accomplished by proving finite bounds on the quadratic, the Hellinger, and the Kullback-Leibler loss of the MDL learner, which are, however, exponentially worse than for Bayesian prediction. We demonstrate that these bounds are sharp, even for model classes containing only Bernoulli distributions. We show how these bounds imply regret bounds for arbitrary loss functions. Our results apply to a wide range of setups, namely, sequence prediction, pattern classification, regression, and universal induction in the sense of algorithmic information theory among others.