We consider the question of Gaussian mean testing, a fundamental task in high-dimensional distribution testing and signal processing, subject to adversarial corruptions of the samples. We focus on the relative power of different adversaries, and show that, in contrast to the common wisdom in robust statistics, there exists a strict separation between adaptive adversaries (strong contamination) and oblivious ones (weak contamination) for this task. Specifically, we resolve both the information-theoretic and computational landscapes for robust mean testing. In the exponential-time setting, we establish the tight sample complexity of testing $\mathcal{N}(0,I)$ against $\mathcal{N}(\alpha v, I)$, where $\|v\|_2 = 1$, with an $\varepsilon$-fraction of adversarial corruptions, to be \[ \tilde{\Theta}\!\left(\max\left(\frac{\sqrt{d}}{\alpha^2}, \frac{d\varepsilon^3}{\alpha^4},\min\left(\frac{d^{2/3}\varepsilon^{2/3}}{\alpha^{8/3}}, \frac{d \varepsilon}{\alpha^2}\right)\right) \right) \,, \] while the complexity against adaptive adversaries is \[ \tilde{\Theta}\!\left(\max\left(\frac{\sqrt{d}}{\alpha^2}, \frac{d\varepsilon^2}{\alpha^4} \right)\right) \,, \] which is strictly worse for a large range of vanishing $\varepsilon,\alpha$. To the best of our knowledge, ours is the first separation in sample complexity between the strong and weak contamination models. In the polynomial-time setting, we close a gap in the literature by providing a polynomial-time algorithm against adaptive adversaries achieving the above sample complexity $\tilde{\Theta}(\max({\sqrt{d}}/{\alpha^2}, {d\varepsilon^2}/{\alpha^4} ))$, and a low-degree lower bound (which complements an existing reduction from planted clique) suggesting that all efficient algorithms require this many samples, even in the oblivious-adversary setting.

翻译：我们考虑高斯均值检验问题——高维分布检验与信号处理中的基本任务，在样本受到对抗性污染的场景下展开研究。我们聚焦于不同对抗者的相对能力，并证明：与鲁棒统计领域的普遍认知相反，该任务中适应性对抗者（强污染）与无意识对抗者（弱污染）存在严格分离。具体而言，我们完整解析了鲁棒均值检验的信息论与计算复杂性图景。在指数时间设定下，我们建立了检验 $\mathcal{N}(0,I)$ 与 $\mathcal{N}(\alpha v, I)$（其中 $\|v\|_2 = 1$）且存在 $\varepsilon$ 比例对抗性污染时的精确样本复杂度： \[ \tilde{\Theta}\!\left(\max\left(\frac{\sqrt{d}}{\alpha^2}, \frac{d\varepsilon^3}{\alpha^4},\min\left(\frac{d^{2/3}\varepsilon^{2/3}}{\alpha^{8/3}}, \frac{d \varepsilon}{\alpha^2}\right)\right) \right) \,, \] 而针对适应性对抗者的复杂度为： \[ \tilde{\Theta}\!\left(\max\left(\frac{\sqrt{d}}{\alpha^2}, \frac{d\varepsilon^2}{\alpha^4} \right)\right) \,, \] 在 $\varepsilon,\alpha$ 趋于零的大范围区域内，后者严格劣于前者。据我们所知，这是强污染与弱污染模型在样本复杂度上的首次分离。在多项式时间设定下，我们弥补了文献中的空白：提供了针对适应性对抗者的多项式时间算法，其达到上述样本复杂度 $\tilde{\Theta}(\max({\sqrt{d}}/{\alpha^2}, {d\varepsilon^2}/{\alpha^4} ))$；同时给出低度下界（补充了基于植团问题的现有归约），表明即使在无意识对抗者设定下，所有高效算法仍需同等数量的样本。