Best Arm Identification (BAI) problems are progressively used for data-sensitive applications, such as designing adaptive clinical trials, tuning hyper-parameters, and conducting user studies to name a few. Motivated by the data privacy concerns invoked by these applications, we study the problem of BAI with fixed confidence under $\epsilon$-global Differential Privacy (DP). First, to quantify the cost of privacy, we derive a lower bound on the sample complexity of any $\delta$-correct BAI algorithm satisfying $\epsilon$-global DP. Our lower bound suggests the existence of two privacy regimes depending on the privacy budget $\epsilon$. In the high-privacy regime (small $\epsilon$), the hardness depends on a coupled effect of privacy and a novel information-theoretic quantity, called the Total Variation Characteristic Time. In the low-privacy regime (large $\epsilon$), the sample complexity lower bound reduces to the classical non-private lower bound. Second, we propose AdaP-TT, an $\epsilon$-global DP variant of the Top Two algorithm. AdaP-TT runs in arm-dependent adaptive episodes and adds Laplace noise to ensure a good privacy-utility trade-off. We derive an asymptotic upper bound on the sample complexity of AdaP-TT that matches with the lower bound up to multiplicative constants in the high-privacy regime. Finally, we provide an experimental analysis of AdaP-TT that validates our theoretical results.

翻译：最佳臂识别（BAI）问题逐渐被应用于数据敏感场景，例如设计自适应临床试验、调优超参数以及开展用户研究等。受这些应用引发的数据隐私担忧的驱动，我们研究了在$\epsilon$-全局差分隐私（DP）约束下具有固定置信度的BAI问题。首先，为量化隐私成本，我们推导了满足$\epsilon$-全局DP的任何$\delta$-正确BAI算法样本复杂度的下界。该下界表明存在两个依赖于隐私预算$\epsilon$的隐私区间：在高隐私区间（小$\epsilon$）中，难度取决于隐私与一种新型信息论量——总变分特征时间——的耦合效应；在低隐私区间（大$\epsilon$）中，样本复杂度下界退化为经典的非隐私下界。其次，我们提出AdaP-TT算法，作为Top Two算法的$\epsilon$-全局DP变体。该算法以臂相关的自适应回合运行，并添加拉普拉斯噪声以确保良好的隐私-效用权衡。我们推导了AdaP-TT样本复杂度的渐近上界，该上界在高隐私区间中与下界匹配至乘法常数项。最后，我们通过实验分析验证了AdaP-TT的理论结果。