Using non-asymptotic error bounds to obtain tight sample complexity for binary testing

Determine whether existing non-asymptotic bounds on Type-I and Type-II error probabilities can be used to derive tight non-asymptotic sample complexity bounds for simple binary hypothesis testing.

Background

The information theory literature provides many non-asymptotic bounds on error probabilities, often developed for channel coding and hypothesis testing contexts. However, these bounds typically do not tensorize in a way that directly yields sharp sample complexity characterizations for i.i.d. product distributions.

This paper introduces a one-shot lower bound with a crucial tensorization property to derive tight sample complexity results, but it explicitly notes that it remains unclear whether the standard non-asymptotic Type-I/Type-II error bounds can lead to similarly tight sample complexity formulas.