Tight Non-asymptotic Inference via Sub-Gaussian Intrinsic Moment Norm
This work addresses a fundamental statistical challenge in machine learning, offering tighter bounds for applications like reinforcement learning, though it builds on existing theoretical frameworks.
The paper tackles the problem of estimating variance-type parameters for sub-Gaussian distributions in non-asymptotic learning, proposing the use of the sub-Gaussian intrinsic moment norm to provide tighter concentration inequalities and a practical method for assessing sub-Gaussianity with finite samples.
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.