A Finite-Sample Strong Converse for Binary Hypothesis Testing via (Reverse) R\'enyi Divergence

TL;DR

This paper establishes a finite-sample strong converse for binary hypothesis testing using reverse Rènyi divergence, providing sharp bounds on error probabilities and revealing phase transitions based on error decay rates.

Contribution

It introduces novel non-asymptotic bounds on Type II error using reverse Rènyi divergence, improving understanding of finite-sample hypothesis testing limits.

Findings

Type II error converges to 1 exponentially fast if error decay rate exceeds divergence

Type II error vanishes exponentially if decay rate is below divergence

Numerical examples show bounds improve upon existing results

Abstract

This work investigates binary hypothesis testing between $H_0\sim P_0$ and $H_1\sim P_1$ in the finite-sample regime under asymmetric error constraints. By employing the ``reverse" R\'enyi divergence, we derive novel non-asymptotic bounds on the Type II error probability which naturally establish a strong converse result. Furthermore, when the Type I error is constrained to decay exponentially with a rate $c$, we show that the Type II error converges to 1 exponentially fast if $c$ exceeds the Kullback-Leibler divergence $D(P_1\|P_0)$, and vanishes exponentially fast if $c$ is smaller. Finally, we present numerical examples demonstrating that the proposed converse bounds strictly improve upon existing finite-sample results in the literature.