Tight Non-asymptotic Inference via Sub-Gaussian Intrinsic Moment Norm

TL;DR

This paper introduces a new method for non-asymptotic inference in sub-Gaussian distributions using the intrinsic moment norm, enabling tighter concentration bounds and practical sub-Gaussian assessment.

Contribution

It proposes utilizing the sub-Gaussian intrinsic moment norm for improved variance parameter estimation and concentration inequalities, with applications to reinforcement learning.

Findings

Provides a robust estimation method for the intrinsic moment norm.

Offers a practical sub-Gaussian assessment tool via the sub-Gaussian plot.

Demonstrates applicability to reinforcement learning scenarios.

Abstract

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.