On the Convergence Speed of MDL Predictions for Bernoulli Sequences

Details On the Convergence Speed of MDL Predictions for Bernoulli Sequences On the Convergence Speed of MDL Predictions for Bernoulli Sequences

2004-07-16

arxiv.org/abs/cs/0407039v1

Proc. 15th International Conf. on Algorithmic Learning Theory (ALT-2004), pages 294-308

Computer Science

Learning

17 pages

Scientific paper

We consider the Minimum Description Length principle for online sequence prediction. If the underlying model class is discrete, then the total expected square loss is a particularly interesting performance measure: (a) this quantity is bounded, implying convergence with probability one, and (b) it additionally specifies a `rate of convergence'. Generally, for MDL only exponential loss bounds hold, as opposed to the linear bounds for a Bayes mixture. We show that this is even the case if the model class contains only Bernoulli distributions. We derive a new upper bound on the prediction error for countable Bernoulli classes. This implies a small bound (comparable to the one for Bayes mixtures) for certain important model classes. The results apply to many Machine Learning tasks including classification and hypothesis testing. We provide arguments that our theorems generalize to countable classes of i.i.d. models.

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