1. Hardness as lower bounds: We commence our study by deriving the lower bound on sample complexity (or expected stopping time) of any FC-BAI algorithm ensuring  $\epsilon$ -global DP, in brief  $\epsilon$ -DP-FC-BAI (Section 3). In Theorem 2, we prove that the sample complexity of  $\epsilon$ -DP-FC-BAI depends on the minimum of two information-theoretic quantities one that depends on privacy, and the other that originates from the classical sample complexity of non-private FC-BAI [KCG16]. The term dependent on privacy depends on the privacy budget  $\epsilon$ , and a novel information-theoretic quantity, referred to as the Total Variation Characteristic Time  $T_{\mathrm{TV}}^{*}$ .  $T_{\mathrm{TV}}^{*}$  depends on the Total Variation (TV) distance between the reward distributions in the bandit problem and corresponding most confusing instance. The lower bound also indicates that, in a similar spirit as the regret minimisation with  $\epsilon$ -global DP [AB22], there are two regimes of hardness for  $\epsilon$ -DP-FC-BAI. For lower level of privacy (i.e. higher  $\epsilon$ ), the sample complexity of  $\epsilon$ -DP-FC-BAI is identical to that of the non-private FC-BAI. But for higher level of privacy (i.e. lower  $\epsilon$ ), the sample complexity depends on  $\epsilon$  and  $T_{\mathrm{TV}}^{*}$ .
2. Algorithm design: Following the lower bounds, we aim to design an efficient  $\epsilon$ -DP-FC-BAI algorithm that simultaneously achieves the lower bound order-optimally and is computationally efficient (Section 4). Due to the superior empirical performance and computational efficiency of Top Two algorithms, we design an  $\epsilon$ -DP variant of Top Two algorithms, named AdaP-TT. Specifically, we show two simple design techniques, i.e. adaptive episodes for each arm and Laplacian mechanism, if properly used, lead to AdaP-TT from the non-private TTUCB [JD22]. We further derive an asymptotic (as  $\delta \to 0$ ) upper bound on the sample complexity of AdaP-TT (Theorem 5). In the high-privacy regime, the sample complexity upper bound of AdaP-TT coincides with the lower bound up to multiplicative constants. Thus, it is an order-optimal  $\epsilon$ -DP-FC-BAI algorithm in this regime, whereas the looseness in the low-privacy regime is as same as the looseness of the non-private Top Two algorithm. In Section 5, we experimentally show that AdaP-TT is more sample efficient than the existing  $\epsilon$ -DP-FC-BAI algorithm, i.e. DP-SE [SS19]. We also show that its sample complexity is independent of the privacy budget in the low-privacy regime, as already indicated by the lower bound.
3. Technical tools: (a) To derive the lower bound, we provide an  $\epsilon$ -global DP version of the transportation lemma [KK13] (Lemma 1), which we prove using a sequential coupling argument. We also define and study the TV characteristic time  $(T_{\mathrm{TV}}^{\star})$  quantifying the hardness of FC-BAI in high privacy regimes (Proposition 1). (b) To design the algorithm, we propose a generic wrapper, which adapts the existing FC-BAI algorithm to tackle DP-FC-BAI. It builds on two components: (i) Adaptive episodes with per-arm doubling and forgetting, (ii) A private GLR stopping rule obtained by plugging in private empirical means in the non-private GLR stopping rule used of FC-BAI with Gaussian distributions. To use this proposed wrapper, one can choose among the numerous existing sampling rules to tackle FC-BAI. In this work, we consider the Top Two algorithms since they have good theoretical guarantees and empirical performance. To provide the sample complexity upper bound, we study step-by-step the effects of doubling, forgetting, and adding noise on the performance of the algorithm. Building on  $[\mathrm{JDB}^{+}22]$ , we provide a generic analysis of the class of Top Two algorithms when combined with our wrapper.