  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.
