In non-asymptotic learning, variance-type parameters of sub-Gaussian distributions are of paramount importance. However, directly estimating these parameters using the empirical moment generating function (MGF) is infeasible. To address this, we suggest using the sub-Gaussian intrinsic moment norm [Buldygin and Kozachenko (2000), Theorem 1.3] achieved by maximizing a sequence of normalized moments. Significantly, the suggested norm can not only reconstruct the exponential moment bounds of MGFs but also provide tighter sub-Gaussian concentration inequalities. In practice, we provide an intuitive method for assessing whether data with a finite sample size is sub-Gaussian, utilizing the sub-Gaussian plot. The intrinsic moment norm can be robustly estimated via a simple plug-in approach. Our theoretical findings are also applicable to reinforcement learning, including the multi-armed bandit scenario.

翻译：在非渐近学习中，次高斯分布的方差型参数具有至关重要的意义。然而，直接使用经验矩母函数（MGF）估计这些参数是不可行的。为解决此问题，我们建议采用通过最大化一系列归一化矩所获得的次高斯内蕴矩范数[Buldygin and Kozachenko (2000), Theorem 1.3]。值得注意的是，所提出的范数不仅能重构矩母函数的指数矩界，还能提供更严格的次高斯集中不等式。在实践中，我们提出一种利用次高斯图的直观方法，用于评估有限样本数据是否具有次高斯性。该内蕴矩范数可通过简单的插件方法进行稳健估计。我们的理论发现同样适用于强化学习，包括多臂老虎机场景。