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

An $\varepsilon$-Best-Arm Identification Algorithm for Fixed-Confidence and Beyond