Universal Batch Learning Under The Misspecification Setting

In this paper we consider the problem of universal {\em batch} learning in a misspecification setting with log-loss. In this setting the hypothesis class is a set of models $\Theta$. However, the data is generated by an unknown distribution that may not belong to this set but comes from a larger set of models $\Phi \supset \Theta$. Given a training sample, a universal learner is requested to predict a probability distribution for the next outcome and a log-loss is incurred. The universal learner performance is measured by the regret relative to the best hypothesis matching the data, chosen from $\Theta$. Utilizing the minimax theorem and information theoretical tools, we derive the optimal universal learner, a mixture over the set of the data generating distributions, and get a closed form expression for the min-max regret. We show that this regret can be considered as a constrained version of the conditional capacity between the data and its generating distributions set. We present tight bounds for this min-max regret, implying that the complexity of the problem is dominated by the richness of the hypotheses models $\Theta$ and not by the data generating distributions set $\Phi$. We develop an extension to the Arimoto-Blahut algorithm for numerical evaluation of the regret and its capacity achieving prior distribution. We demonstrate our results for the case where the observations come from a $K$-parameters multinomial distributions while the hypothesis class $\Theta$ is only a subset of this family of distributions.