We call that σ -algebra history before n . Identification tasks We focus on best arm identification (BAI).

Recall (higher is better): the ratio of selected ID instances to the total number of ID instances.

Recall (higher is better): the ratio of selected ID instances to the total number of ID instances.

Recall (higher is better): the ratio of selected ID instances to the total number of ID instances.

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.