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.

翻译：受通用预测中经典遗憾度量与Rényi散度之间联系的启发，我们引入了一类依赖于实参数$\alpha\geq 1$的新型通用预测器。该类预测器内插了两种广为人知的预测器——混合估计器（包括拉普拉斯预测器和克雷切夫斯基-特罗菲莫夫预测器）以及归一化最大似然（NML）估计器。我们指出了这类新预测器的若干优势，并从两个互补视角研究了其益处：（1）当将最大Rényi散度作为遗憾度量时，我们证明其最优性——这在操作意义上可解释为介于标准平均遗憾与最坏情况遗憾度量之间的中间地带；（2）我们讨论了在NML不可行时如何将其作为幸运NML等其他预测器的替代方案。最后，我们将$\alpha$-NML预测器应用于离散无记忆信源（DMS）类，推导出计算该预测器的简易公式，并基于最坏情况遗憾分析了其渐近性能。