On the Sample Complexity of Robust Binary Hypothesis Testing

2026-05-23

AI Summary

This study investigates the sample complexity and sensitivity to contamination parameters in robust binary hypothesis testing under ε-additive (Huber), ε-subtractive, and ε-total variation contamination models. By leveraging information-theoretic and statistical decision-theoretic tools—combined with extremal distribution constructions and inequality analysis—the work establishes, for the first time, a least favorable distribution for the subtractive contamination model, revealing instability in sample complexity with respect to the contamination parameter. The main contributions include deriving tight upper and lower bounds on the sample complexity across all three models, demonstrating that these bounds differ only by constant factors, thereby establishing approximate equivalence among the models. Furthermore, the results are extended to adaptive contamination settings, highlighting the critical role of estimation accuracy of contamination parameters in determining sample requirements.

Abstract

We study the sample complexity of robust binary hypothesis testing under three standard contamination models: $\varepsilon$-additive (Huber), $\varepsilon$-subtractive, and $\varepsilon$-total variation (TV), denoted by $n^*_{\mathrm{Hub}}(\varepsilon)$, $n^*_{\mathrm{Sub}}(\varepsilon)$, and $n^*_{\mathrm{TV}}(\varepsilon)$, respectively. For subtractive contamination, we show that least favourable distributions exist and provide explicit formulas for the same, bringing this model in line with the classical Huber and TV models. Next we show that in all three models, sample complexity may be highly unstable in the contamination parameter $\varepsilon$, increasing by polynomial factors even for $o(\varepsilon)$ perturbations. Similarly, there may be polynomial factor gaps between the sample complexities when $\varepsilon$ is known exactly versus when it is known up to $o(\varepsilon)$ error. Despite the instability of the sample complexity in all models, we show that the sample complexities across models are comparable up to constant-factor rescaling of $\varepsilon$. Specifically, for any fixed $δ_0>0$, the following hold for all distributions $p$ and $q$: (i) $n^*_{\mathrm{Hub}}(\varepsilon) \lesssim n^*_{\mathrm{TV}}(\varepsilon) \lesssim n^*_{\mathrm{Hub}}(2\varepsilon)$, (ii) $n^*_{\mathrm{Sub}}(\varepsilon) \lesssim n^*_{\mathrm{TV}}(\varepsilon) \lesssim n^*_{\mathrm{Sub}}((2+δ_0)\varepsilon)$, and (iii) $n^*_{\mathrm{Sub}}(\varepsilon) \lesssim n^*_{\mathrm{Hub}}(\varepsilon) \lesssim n^*_{\mathrm{Sub}}((1+δ_0)\varepsilon)$, and the scaling constants are tight. Finally, we extend our results to adaptive versions of the contamination models.

Problem

• sample complexity
• robust hypothesis testing
• contamination models
• binary hypothesis testing
• total variation

Innovation

• sample complexity
• robust hypothesis testing
• contamination models
• least favorable distributions
• adaptive robustness