基于Rényi散度的有限样本二元假设检验：强逆命题与局部隐私
Finite-Sample Binary Hypothesis Testing via Rényi Divergences: Strong Converse and Local Privacy
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中文总结 AI 辅助
本文通过Rényi散度的变分表示，为二元假设检验建立有限样本强逆命题，统一恢复已知逆界，并刻画隐私下的相变与样本复杂度。
中文摘要 AI 辅助
我们研究了基于$n$个独立同分布观测的$H_0:P_0^{n}$与$H_1:P_1^{n}$之间的非对称简单二元假设检验问题。利用阶数为$\alpha$的Rényi散度的变分表示，我们推导出主要结果：一个$\alpha>1$时的有限样本逆命题。该界同时使用散度$D_\alpha(P_1\\|P_0)$和$D_\alpha(P_0\\|P_1)$的两个方向，在乘积测度下满足张量化性质，并将常见的数据处理逆命题作为边界情况包含在内。为了比较，我们将相同的变分方法应用于一般的$f$-散度，并专门化到全变差、$E_\gamma$、Hellinger和Kullback-Leibler散度，从而在统一框架内恢复熟悉的逆命题。结合涉及$\alpha\in(0,1)$的Rényi散度的可达性界，主要逆命题在指数衰减的I型错误约束$\varepsilon_n=e^{-nr}$下恢复了最优II型错误的相变。在正则条件下，当$r<D(P_1\\|P_0)$时最优II型错误指数级消失，当$r>D(P_1\\|P_0)$时指数级收敛到1。我们还推导了样本复杂度界，并将逆命题和可达性分析扩展到局部差分隐私观测，量化了隐私代价，并在隐私约束消失时恢复非隐私可达性界。
英文摘要
We study asymmetric simple binary hypothesis testing between $H_0:P_0^{n}$ and $H_1:P_1^{n}$, based on $n$ independent and identically distributed observations. Leveraging a variational representation of Rényi divergence of order $α$, we derive our main result: a finite-sample converse with $α>1$. The bound uses both directions of the divergence $D_α(P_1\|P_0)$ and $D_α(P_0\|P_1)$, tensorises under product measures, and contains familiar data-processing converses as boundary cases. For comparison, we apply the same variational approach to general $f$-divergences and specialise it to total variation, $E_γ$, Hellinger, and Kullback Leibler divergences, thereby recovering familiar converses within a unified framework. Together with an achievability bound involving Rényi divergence with $α\in (0,1)$, the main converse recovers the phase transition of the optimal Type II error under the exponentially decaying Type I error constraint $\varepsilon_n=e^{-nr}$. Under regularity conditions, the optimal Type II error vanishes exponentially when $r<D(P_1\|P_0)$ and converges exponentially fast to one when $r>D(P_1\|P_0)$. We also derive sample-complexity bounds and extend both the converse and achievability analyses to locally differentially private observations, quantifying the cost of privacy and recovering the non-private achievability bound as the privacy constraint vanishes.
发表机构
- University of Salerno（萨莱诺大学）
- EPFL（洛桑联邦理工学院）
- Okinawa Institute of Science and Technology（冲绳科学技术大学院大学）
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