In differential privacy (DP), the generalized private testing problem was introduced by Liu and Talwar (STOC 2019). Given a dataset $X \in \mathcal{X}$ and a sequence of black-box $\varepsilon_t$-DP mechanisms $M_t:\mathcal{X}\to\{+1,-1\}$, the analyst must accept the first mechanism whose success probability $p_t=\Pr[M_t(X)=+1]$ exceeds a given threshold $p^*\in(0,1)$, while achieving DP. Accuracy is measured by the gap between $p^*$ and a rejection threshold $\bar{p}$, such that with probability $1-β$ for all $t\geq1$, if $p_t\leq\bar{p}$, then $M_t$ is rejected, and if $p_t\geq p^*$, then it is accepted. This generalizes the standard private testing problem, whose solution, the Sparse Vector Technique, is ubiquitous in DP. We introduce the Generalized Thresholding Mechanism (GTM) for generalized private testing. For $\varepsilon>0$ and any sequence of $(\varepsilon_t,δ_t)$-DP mechanisms $M_t$, the GTM is pure $\varepsilon$-DP. For $θ>0$, $γ\in(1,2]$, and $β\in(0,1)$, $\bar{p}_t=\max(p^*/γΛ_t, 1 - γΛ_t(1-p^*))-δ_t/\varepsilon_t$ for $Λ_t=(5t\ln^3(t+2))^{(2+θ)\varepsilon_t/\varepsilon}(4/β)^{(3+θ+2/θ)\varepsilon_t/\varepsilon}$. With probability $1-β$, the number of evaluations of $M_t$ is at most $O((\ln(t/β)/(γ-1)^2)\max(Λ_t/p^*,(1-p^*)^{-1}))$ for all $t\geq 1$. Our lower bounds prove near-optimality of our accuracy and sample complexity guarantees. Via the GTM, we give a black-box reduction for DP optimization from the continual observation (CO) setting to the batch setting. This gives us the first DP-CO algorithms for many maximization problems. Further, the GTM permits an adaptive choice of acceptance thresholds $(p^*_t)_{t\geq1}$, addressing a challenge mentioned in prior work on using generalized private testing for hyperparameter optimization (Papernot and Steinke (ICLR 2022)).

翻译：在差分隐私（DP）中，广义私有测试问题由Liu和Talwar（STOC 2019）提出。给定数据集 $X \in \mathcal{X}$ 和一系列黑盒 $\varepsilon_t$-DP 机制 $M_t:\mathcal{X}\to\{+1,-1\}$，分析者必须接受首个成功概率 $p_t=\Pr[M_t(X)=+1]$ 超过给定阈值 $p^*\in(0,1)$ 的机制，同时实现DP。准确性通过 $p^*$ 与拒绝阈值 $\bar{p}$ 之间的差距来衡量，使得对于所有 $t\geq1$ 以概率 $1-β$ 满足：若 $p_t\leq\bar{p}$，则拒绝 $M_t$；若 $p_t\geq p^*$，则接受 $M_t$。这推广了标准私有测试问题，其解决方案——稀疏向量技术——在DP中无处不在。我们为广义私有测试引入了广义阈值机制（GTM）。对于 $\varepsilon>0$ 和任意序列的 $(\varepsilon_t,δ_t)$-DP 机制 $M_t$，GTM是纯 $\varepsilon$-DP的。对于 $θ>0$，$γ\in(1,2]$ 和 $β\in(0,1)$，$\bar{p}_t=\max(p^*/γΛ_t, 1 - γΛ_t(1-p^*))-δ_t/\varepsilon_t$，其中 $Λ_t=(5t\ln^3(t+2))^{(2+θ)\varepsilon_t/\varepsilon}(4/β)^{(3+θ+2/θ)\varepsilon_t/\varepsilon}$。以概率 $1-β$，对于所有 $t\geq 1$，对 $M_t$ 的评估次数不超过 $O((\ln(t/β)/(γ-1)^2)\max(Λ_t/p^*,(1-p^*)^{-1}))$。我们的下界证明了准确性和样本复杂度保证的近乎最优性。通过GTM，我们给出了从持续观察（CO）场景到批处理场景的DP优化黑盒归约。这为我们首次提供了许多最大化问题的DP-CO算法。此外，GTM允许自适应选择接受阈值 $(p^*_t)_{t\geq1}$，解决了先前工作（Papernot和Steinke，ICLR 2022）中关于使用广义私有测试进行超参数优化时提到的挑战。