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

Marc Jourdan, Rémy Degenne, Emilie Kaufmann

Keywords: multi-armed bandits, pure-exploration, epsilon best arm identification, Top Two algorithm, anytime

TL;DR: We have studied approximate best arm identification in multi-armed bandits, in the fixed-confidence, fixed-budget and anytime settings.

Abstract: 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 slack 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 for different approximate best arm identification tasks.