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$\varepsilon$，一种用于随机多臂赌博机中$\varepsilon$-最优臂识别的新型采样规则。这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*随时*采样规则，因此无需修改即可直接用于固定置信度或固定预算（无需预先知晓预算）的识别任务。我们为EB-TC$\varepsilon$提供了三类理论保证。首先，在固定置信度场景下，我们证明了其期望样本复杂度的上界，尤其表明当其探索参数经过自适应调整时可达到渐近最优性。我们通过任意时刻及任意误差参数下的错误概率上界来补充这些发现，这进一步推导出任意时刻简单遗憾的上界。最后，通过数值模拟表明，在不同场景下EB-TC$\varepsilon$的性能均优于现有算法。