Asymptotics of Discrete MDL for Online Prediction
Abstract
Minimum Description Length (MDL) is an important principle for induction and prediction, with strong relations to optimal Bayesian learning. This paper deals with learning noni.i.d. processes by means of twopart 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 his 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 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2005
 arXiv:
 arXiv:cs/0506022
 Bibcode:
 2005cs........6022P
 Keywords:

 Computer Science  Information Theory;
 Computer Science  Machine Learning;
 Mathematics  Statistics;
 I.2.6;
 E.4;
 G.3
 EPrint:
 34 pages