Alpha-NML Universal Predictors

Abstract

Inspired by the connection between classical regret measures employed in universal prediction and R\'{e}nyi divergence, we introduce a new class of universal predictors that depend on a real parameter $\alpha\geq 1$. This class interpolates two well-known predictors, the mixture estimators, that include the Laplace and the Krichevsky-Trofimov predictors, and the Normalized Maximum Likelihood (NML) estimator. We point out some advantages of this new class of predictors and study its benefits from two complementary viewpoints: (1) we prove its optimality when the maximal R\'{e}nyi divergence is considered as a regret measure, which can be interpreted operationally as a middle ground between the standard average and worst-case regret measures; (2) we discuss how it can be employed when NML is not a viable option, as an alternative to other predictors such as Luckiness NML. Finally, we apply the $\alpha$-NML predictor to the class of discrete memoryless sources (DMS), where we derive simple formulas to compute the predictor and analyze its asymptotic performance in terms of worst-case regret.