Algorithmic Learning Theory

Volume 3244 of the series Lecture Notes in Computer Science pp 294-308

On the Convergence Speed of MDL Predictions for Bernoulli Sequences

  • Jan PolandAffiliated withIDSIA
  • , Marcus HutterAffiliated withIDSIA

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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.