We propose EB-TC$\varepsilon$, a novel sampling rule for $\varepsilon$-best arm identification in stochastic bandits. It is the first instance of Top Two algorithm analyzed for approximate best arm identification. EB-TC$\varepsilon$ is an *anytime* sampling rule that can therefore be employed without modification for fixed confidence or fixed budget identification (without prior knowledge of the budget). We provide three types of theoretical guarantees for EB-TC$\varepsilon$. First, we prove bounds on its expected sample complexity in the fixed confidence setting, notably showing its asymptotic optimality in combination with an adaptive tuning of its exploration parameter. We complement these findings with upper bounds on its probability of error at any time and for any error parameter, which further yield upper bounds on its simple regret at any time. Finally, we show through numerical simulations that EB-TC$\varepsilon$ performs favorably compared to existing algorithms, in different settings.

翻译：我们提出EB-TC$\varepsilon$，这是一种用于随机赌博机中$\varepsilon$-最优臂识别的新型采样规则。它是首个针对近似最优臂识别问题进行分析的Top Two算法。EB-TC$\varepsilon$是一种*任意时刻*采样规则，因此无需修改即可用于固定置信度或固定预算场景下的臂识别（无需事先了解预算信息）。我们为EB-TC$\varepsilon$提供了三类理论保证。首先，我们证明了其在固定置信度场景下的期望样本复杂度上界，特别地，通过自适应调整探索参数，该算法具有渐近最优性。我们通过任意时刻及任意误差参数下的错误概率上界对这些发现进行补充，这些上界进一步推导出任意时刻下简单遗憾的上界。最后，数值仿真表明，在不同设定下EB-TC$\varepsilon$相较于现有算法具有更优性能。