We study best arm identification (BAI) in linear bandits in the fixed-budget regime under differential privacy constraints, when the arm rewards are supported on the unit interval. Given a finite budget $T$ and a privacy parameter $\varepsilon>0$, the goal is to minimise the error probability in finding the arm with the largest mean after $T$ sampling rounds, subject to the constraint that the policy of the decision maker satisfies a certain {\em $\varepsilon$-differential privacy} ($\varepsilon$-DP) constraint. We construct a policy satisfying the $\varepsilon$-DP constraint (called {\sc DP-BAI}) by proposing the principle of {\em maximum absolute determinants}, and derive an upper bound on its error probability. Furthermore, we derive a minimax lower bound on the error probability, and demonstrate that the lower and the upper bounds decay exponentially in $T$, with exponents in the two bounds matching order-wise in (a) the sub-optimality gaps of the arms, (b) $\varepsilon$, and (c) the problem complexity that is expressible as the sum of two terms, one characterising the complexity of standard fixed-budget BAI (without privacy constraints), and the other accounting for the $\varepsilon$-DP constraint. Additionally, we present some auxiliary results that contribute to the derivation of the lower bound on the error probability. These results, we posit, may be of independent interest and could prove instrumental in proving lower bounds on error probabilities in several other bandit problems. Whereas prior works provide results for BAI in the fixed-budget regime without privacy constraints or in the fixed-confidence regime with privacy constraints, our work fills the gap in the literature by providing the results for BAI in the fixed-budget regime under the $\varepsilon$-DP constraint.

在差分隐私约束下，研究固定预算探索期的线性赌臂问题，通过最大绝对确定性原则构建满足差分隐私约束的策略，得到其错误概率的上限和下限，并展示其与赌臂问题复杂性、亚最优的臂差异和差分隐私参数相关的指数级衰减特性。此外，该研究还提供了独立感兴趣且对其他赌臂问题的错误概率下限证明具有重要作用的一些辅助结果，填补了先前文献中固定预算探索期下差分隐私约束情况下最优臂识别的研究空白。

固定预算差分私有最佳臂辨识