On the Complexity of Differentially Private Best-Arm Identification with Fixed Confidence

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 ε-global Differential Privacy (DP). First, to quantify the cost of privacy, we derive a lower bound on the sample complexity of any δ-correct BAI algorithm satisfying ε-global DP. Our lower bound suggests the existence of two privacy regimes depending on the privacy budget ε. In the high-privacy regime (small ε), 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 ε), the sample complexity lower bound reduces to the classical non-private lower bound. Second, we propose AdaP-TT, an ε-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.