Asymptotics of discrete MDL for online prediction

• Published in: IEEE transactions on information theory Vol. 51; no. 11; pp. 3780 - 3795
• Main Authors: Poland, J., Hutter, M.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.11.2005; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithmic information theory; Algorithms; Applied sciences; Asymptotic properties; Bayesian analysis; Bayesian methods; classification; consistency; Convergence; discrete model class; Distribution; Exact sciences and technology; Hybrid power systems; Information science; Information systems; Information theory; Information, signal and communications theory; Learning; loss bounds; Materials science and technology; Mathematical models; minimum description length (MDL); On-line systems; Online; Pattern classification; Pattern recognition; Predictive models; Regression; sequence prediction; stabilization; Statistical learning; Telecommunications and information theory; universal induction; Bayes estimation; sequence prediction; Predictor; Stabilization; On-line systems; classification; consistency; Learning; discrete model class; Loss function; minimum description length (MDL); Pattern classification; regression; MDL criterion; universal induction; Information theory; Algorithmic information theory; loss bounds
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2005.856956

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.