AI 中文总结
研究在中心差分隐私下Hölder光滑密度的两样本检验，通过离散化样本、应用私有离散检验等方法给出上界，结合多种构造和不等式证明下界，未知光滑度时开发多尺度私有检验并证明匹配下界，得到尖锐极小极大率和最优自适应率。
AI 中文摘要
我们建立了在中心差分隐私下Hölder光滑密度的两样本检验的尖锐极小极大极限。给定两个独立样本，目标是判定基础分布在\(L_1\)距离上是否相同或分离，同时仅释放一个\(\varepsilon -\)差分隐私决策。我们表明隐私通过多种情况改变经典光滑检验边界：最优分离半径是四项的最大值，包括经典非隐私率和三个不同的隐私诱导障碍。哪个障碍起作用取决于隐私预算和平滑度与维度比，产生一个尖锐的相图。我们的上界对样本进行离散化，对所得直方图应用一个私有离散两样本检验，并选择箱分辨率以平衡近似偏差、采样波动和隐私噪声。该过程还允许具有有限样本I型错误控制的置换校准实现。对于下界，我们将光滑扰动构造与特定于隐私的耦合和运输不等式相结合，表明所有四项都是不可避免的。最后，当光滑度未知时，我们开发了一个多尺度私有检验，它达到最优自适应率并证明了一个匹配的下界。自适应恰好花费一个迭代对数因子，并且这个代价仅出现在经典非隐私项中。
英文摘要
We establish sharp minimax limits for two-sample testing of Hölder-smooth densities under central differential privacy. Given two independent samples, the goal is to decide whether the underlying distributions are identical or separated in $L_1$ distance, while releasing only an $\varepsilon$-differentially private decision. We show that privacy changes the classical smooth-testing boundary through multiple regimes: the optimal separation radius is the maximum of four terms, consisting of the classical nonprivate rate and three distinct privacy-induced barriers. Which barrier is active depends on the privacy budget and the smoothness-to-dimension ratio, yielding a sharp phase diagram. Our upper bound discretizes the samples, applies a private discrete two-sample test to the resulting histograms, and chooses the bin resolution to balance approximation bias, sampling fluctuations, and privacy noise. The procedure also admits a permutation-calibrated implementation with finite-sample type~I error control. For the lower bounds, we combine smooth perturbation constructions with privacy-specific coupling and transport inequalities, showing that all four terms are unavoidable. Finally, when the smoothness is unknown, we develop a multiscale private test that attains the optimal adaptive rate and prove a matching lower bound. Adaptation costs exactly an iterated-logarithmic factor, and this cost appears only in the classical nonprivate term.
